Weak quasi-Hopf algebras, C*-tensor categories and conformal field theory, and the Kazhdan-Lusztig-Finkelberg theorem
Sebastiano Carpi, Sergio Ciamprone, Marco Valerio Giannone, Claudia Pinzari
aa r X i v : . [ m a t h . QA ] F e b WEAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES ANDCONFORMAL FIELD THEORY, AND ON AN APPROACH TOKAZHDAN-LUSZTIG-FINKELBERG THEOREM
SEBASTIANO CARPI, SERGIO CIAMPRONE, MARCO VALERIO GIANNONE,AND CLAUDIA PINZARI
Dedicated to Sergio Doplicher on the occasion of his 80th birthday, to Roberto Longoand to the memory of Vaughan F. R. Jones and John E. Roberts(Preliminary uncompleted unpolished version)
Abstract.
We discuss tensor categories motivated by Conformal Field Theory, theirunitarizability with applications to various models including the affine VOAs. We discussclassification of type A Verlinde fusion categories.We propose an approach to a direct proof of Kazhdan-Lusztig-Finkelberg theorem.This theorem gives a ribbon equivalence between the fusion category associated to aquantum group at a certain root of unity and that associated to a corresponding affinevertex operator algebra at a suitable positive integer level. We develop ideas of a 1998paper by Wenzl.Our results rely on the notion of weak-quasi-Hopf algebra of Drinfeld-Mack-Schomerus.We were also guided by Drinfeld first proof of Drinfeld-Kohno, by the general schemesettled by Bakalov and Kirillov and by Neshveyev and Tuset for a generic parameter butdifferences arise.Wenzl described a fusion tensor product in quantum group fusion categories, and re-lated it to the unitary structure. Given two irreducible objects, the inner product of thefusion tensor product is induced by the braiding of U q ( g ).Moreover, in our interpretation the paper suggests a suitable untwisting procedure bysome square root construction to make the unitary structure trivial. Then it also describesa continuous path that intuitively connects objects of the quantum group fusion categoryto representations of the simple Lie group defining the affine Lie algebra. To approachthis, we study this untwisting procedure.One of our main results is the construction of a Hopf algebra in a weak sense (w-Hopfalgebra) associated to quantum group fusion category and of a twist of it giving a wqhstructure on the Zhu algebra and thus a unitary modular fusion category structure on thecategory of C*-representations of the affine Lie algebra.In particular, the braiding associated to the affine Lie algebra is of a very simple formsimilarly to the case of Drinfeld quasi-Hopf algebra. The associator is a 3-coboundaryin a suitable weak sense. We conjecture that this modular tensor category structureis equivalent to that obtained via the tensor product theory of VOAs by Huang andLepowsky. A proof of our conjecture leads to a proof of Kazhdan-Lusztig-Finkelbergtheorem. We shall try to develop our conjecture in a different paper, or in a later updateof this paper. We next summarize our results in a more precise way. Abstract.
Our main tool is Tannaka-Krein duality for semisimple categories. Afterdeveloping general algebraic theory of weak quasi-Hopf algebras and reviewing the Tan-nakian formalism, we discuss a corresponding analytic theory, which is based on the notionof Ω-involution by Gould and Lekatsas.We introduce the notion of w-Hopf algebra as an analogue of the notion of Hopf algebrain a weak setting. We extend the theory of compact quantum groups in the work byWoronowicz’, and many others.We notice that weak quasi-Hopf algebras may be associated to semisimple tensor cat-egories under very mild assumptions, e.g. amenability, that allow to construct integralvalued submultiplicative dimension functions (weak dimension functions), extending orig-inal results by Mack and Schomerus and Haring-Oldenburg.We use this idea to construct unitary tensor structures on C ∗ -categories that are tensorequivalent to unitary tensor categories. Applications include unitarization of affine VOAs,built on the known tensor equivalence by Kazhdan-Lusztig-Finkelberg-Huang-Lepowskyequivalence and unitarity of quantum group fusion categories by Kirillov-Wenzl-Xu.In particular, we apply our approach to solve a problem posed by Galindo on uniquenessof the unitary tensor structure.In the second part of the paper we study unitary tensor structures of Verlinde fusioncategories more in detail, motivated by the need of a better understanding of whetherour approach to unitarizability of affine VOAs via weak quasi-Hopf algebras is a naturalmanifestation of structural aspects.We classify Verlinde fusion categories of type A , based on Kazhdan-Wenzl theory and onthe w-Hopf algebra previously constructed by the first and last named authors, extendinga result by Bischoff for s l at integer level and Nashveyev and Yamashita for s l N in thegeneric case.Then we approach the connection problem between affine VOAs and quantum groupfusion categories. We follow a scheme indicated by Neshveyev-Tuset-Yamashita for q generic based on the use of discrete quasi-Hopf algebras of Drinfeld, extending it to theweak generalization introduced by Mack and Schomerus, that is we work with discreteweak quasi-Hopf algebras. These weak versions still admit a notion of twist.We generalize the notion of 3-coboundary associator to the weak setting. We introducethe notion of unitary coboundary wqh. In this case the Ω-involution is induced by thebraiding by abstracting the case of U q ( g ). We give a categorical characterization andturns out to extends symmetric tensor functors in Doplicher-Roberts theorem.We formulate an abstract converse of Drinfeld-Kohno theorem in an analytic settingfor a specific subclass providing an untwisted unitary coboundary wqh algebra in thesubclass, that is with the mentioned very simple R -matrix similarly to Drinfeld case andalso a trivial unitary structure.We construct a semisimple unitary coboundary w-Hopf algebra structure on Wenzlalgebra A W (a semisimple subquotient of U q ( g )) with representation category equivalentto the corresponding Verlinde quantum group fusion category. In this case ∆( I ) is givenby Wenzl idempotent P . Subclass membership follows from the w-Hopf property. Weapply our Drinfeld-Kohno to the twist T = R / ∆( I ). In this way we construct a 3-coboundary Drinfeld associator. Finally, we transport an untwisted unitary coboundarycocommutative wqh algebra structure to the Frenkel-Zhu algebra A Z via Wenzl path andfrom this to the corresponding affine Lie algebra representation category that makes itinto a unitary modular fusion category.Possible future directions that we feel interested and we wish to complete in an updatedversion is to resume our approach to a direct proof of Kazhdan-Lusztig-Finkelberg equiv-alence theorem between UMFC categories from quantum groups and affine VOAs startingwith the the tensor product theory by Huang and Lepowsky that is only briefly hinted inthis version. Moreover, we would like to propose to interested people including ourselves todevelop more connections between quantum groups and works in conformal net theory byLongo, Guido-Longo, A. Wassermann approach with the idea of primary fields, Toledano-Laredo work, or on their relation with VOAs by Carpi-Kawahigashi-Longo-Weiner, andGui, or as an analogue of the idea of a compact gauge group by Doplicher and Robertsin high dimensional QFT theory. Any comment is welcome. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 3
Contents
1. Introduction 32. Preliminaries on tensor categories and their functors 73. Rigidity, ribbon category and weak tensor functors 154. Weak quasi-Hopf algebras 185. Tannaka-Krein duality and integral weak dimension functions (wdf) 246. w-Hopf algebras 337. Quasitriangular and ribbon structures 388. Ω-involution and C*-structure 459. The categories Rep h ( A ) and Rep + ( A ) 5010. Unitary braided symmetry and involutive Tannaka-Krein duality 5511. Unitarizability of representations and rigidity 5912. Turning C ∗ -categories into tensor C ∗ -categories, I 6413. Positive wdf and amenability 6914. Constructing integral wdf and uniqueness of unitary tensor structure 7215. Examples of fusion categories with different natural integral wdf 7416. Quantum groups at roots of unity, fusion categories and unitary ribbon wqhalgebras via wdm 7617. VOAs, the Zhu algebra and conformal nets 8018. Kazhdan-Wenzl theory and equivalence of ribbon s l N,q,ℓ -categories 9019. Turning C ∗ -categories into tensor C ∗ -categories, II 10420. Coboundary categories and Deligne’s theorem 10721. Hermitian coboundary wqh algebras 11422. A categorical characterization of discrete hermitian coboundary wqh 12123. Compatible unitary coboundary wqh and an abstract analogue of Drinfeld-Kohno theorem 12324. Compatible unitary coboundary w-Hopf algebras A W ( g , q, ℓ ) as a subquotientof U q ( g ) 12725. Compatible unitary coboundary wqh algebra structure on the Zhu algebra A ( V g k ) as a subquotient of U ( g ), connection with work by CKLW, CWX, FKLequivalence theorem 137References 1381. Introduction
To us, the history of the connection between quantum groups and conformal field theoryor between different appraches to conformal field theories is very fascinating. We shall limitourselves to a few far from complete remarks, more information may be found e.g. in [67],[5], [100] [20], [56], [57] and references to the original papers.
S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI
In 1988 Moore and Seiberg [90], [91] conjectured that chiral rational conformal fieldtheories give rise to tensor categories.In 1989 Drinfeld introduced the notion of quasi-Hopf algebra as a generalization of thenotion of Hopf algebra to the case where the coproduct is not coassociative, and showed thatthe class of quasi-Hopf algebras is closed under an operation, called twist. Categorically,a quasi-Hopf algebra gives a tensor category via its representations, and twist equivalentquasi-Hopf algebras give tensor equivalent tensor categories. Drinfeld used the idea of twistto extend an earlier result of Kohno and showed in this way a deep connection betweenbraiding arising in quantum groups and conformal field theory in the setting of deformationby a formal parameter h . Thus representation theory of these models is described by aquasi-Hopf algebra with classical algebra and coproduct, a simple R -matrix but a highlynon trivial associator associated to the Knizhnik-Zamolodchikov differential equations ofconformal field theory [34].Kazhdan, Lusztig, Finkelberg, considered the case of quantum groups at certain complexroots of unity on one hand and affine Lie algebras at integer levels on the other. Despitesemisimplicity of the categories involved, the quantized universal enveloping algebra of aquantum group at roots of unity has a non-semisimple representation theory. Furthermorerepresentations of affine Lie algebras are infinite dimensional, and the proof of the equiva-lence of the associated semisimple categories becomes substantially more difficult to followas compared to the previous case [39, 40, 77, 84].The theory of vertex operator algebras has reached a very developed state by the work ofFrenkel, Huang, Lepowsky, Zhu and other authors [41, 42, 61, 62, 63, 71]. In particular, theaffine vertex operator algebras have a representation theory that describe WZW models atpositive integer levels, and combination with the previous works gives a deep connectionbetween quantum groups at roots of unity, and affine vertex operator algebras, althoughthis connection seems considered indirect and complicated.The theory of conformal nets originates in the work of Haag and Kastler, Doplicher,Haag and Roberts, in 4-dimensional Algebraic Quantum Field Theory, and importantresults have been obtained by Kawahigashi, Longo, and other authors. Recent work showsa connection between vertex operator algebras and conformal nets under a general setting,first developed in the vacuum representation by Carpi, Kawahigashi, Longo, Weiner andthen extended to representation theory by Gui for many models.Our collaboration originated from the desire to attempt to understand these connec-tions between the three areas, quantum groups and CFT in the setting of vertex operatoralgebras or conformal nets at the level of representation theory. Our approach mainlyfocuses on semisimple tensor categories and construction of quantum groups associatedto them via Tannakian duality, thus we try to compare the theories on a common basiswhich eliminates the nonsemisimle part of quantum groups at roots of unity and infinitedimensionality of modules of vertex operator algebras.We connect with an idea by Mack and Schomerus of the early nineties. They introducedthe notion of weak quasi-Hopf algebra as an extension of that of quasi-Hopf algebra to thecase where the coproduct is not unital. The class allows an analogue of twist deformation. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 5
Apart some early work, to our knowledge very few papers have been dedicated to developthis theory.In particular, it seems to us that an analogue of the notion of Hopf algebra among weakquasi-Hopf algebras is not available, and this seems important to recover Drinfeld’s ideaof 3-coboundary associator in his proof of Drinfeld-Kohno theorem.In this paper we develop a rather complete theory of weak quasi-Hopf algebras over thefield of complex numbers and the Tannakian formalism between them and tensor categories.We discuss extra structure such as quasitriangular and ribbon structures correspondingcategorically to a braiding and a ribbon structure.Then we make a proposal of the weak analogue of Hopf algebras, that we call w-Hopfalgebras, among weak quasi-Hopf algebras in a cohomological interpretation. We alsointroduce a notion of weak tensor functor between tensor categories. A w-Hopf algebra ischaracterised, via Tannaka-Krein duality, by a semisimple rigid tensor category endowedwith a weak tensor functor to Vec.We develop a theory for w-Hopf algebras which includes the notion of 2-cocycle defor-mation, quasi-triangular and ribbon structure. We introduce twisted Hermitian or C*-structures, and study the relationship with the ribbon structure, and with unitary braidedsymmetry and coboundary symmetry for the representation category. In particular, we in-troduce the notion of unitary coboundary w-Hopf C*-algebra. We show that the examplesassociated to the fusion category of U q ( s l N ) at roots of unity as developed in a previouspaper are of this kind.Then we discuss C ∗ (also called unitarity) aspects of the algebras, that we call positiveΩ-involution in a general setting.For a general Ω-involutive weak quasi-bialgebra, the category of ∗ -representations onHermitian spaces turns out to be a tensor ∗ -category, under the fusion tensor productdefined by the Hermitian form associated to the action of Ω. If Ω is positive, the fullsubcategory of Hilbert space representations is a tensor C ∗ -category. We also show therigidity property in either of the three settings if there is an antipode. In the settingof unitary discrete w-Hopf C*-algebras, we make an explicit construction of conjugates,extending a result of [131] for usual C*-involutions.One of our results relying on the use of unitary wqh concerns uniqueness of the unitarystructures in very wide classes of tensor categories solving a problem posed by Galindo in[49]. We were encouraged to further the study of wqh by the illuminating simplicity ofthe proof offered by these algebras that we perceived. We note that closely related resultshave been obtained by Reutter with different methods [109].Then we introduce a subclass of the class of unitary weak quasi-Hopf algebras, that wecall unitary coboundary weak quasi-Hopf algebras . This theory has a twofold motivation.On one side it is motivated both by the notion of coboundary which plays a role in theproof of Drinfeld-Kohno theorem. On the other side, follows closely the study of unitarityof the fusion actegories associated to quantum groups at roots of unity by Wenzl. Forexample, in our terminology, Wenzl showed, among other things, that U q ( g ) is an Hermitiancoboundary Hopf algebra, and used this structure to show a conjecture of Kirillov about S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI positivity of a ∗ -structure on the associated fusion C ( g , q, ℓ ) category for certain ”minimal”roots of unity q . Similar results were obtained by Xu with different methods.H¨aring-Oldenburg extended to the weak case the work of Majid [88, 89] for quasi-Hopfalgebras. He defined a weak quasi-tensor fiber functor from a semisimple rigid tensorcategory and formulated a Tannaka-Krein duality theorem showing that such pairs are induality with semisimple weak quasi-Hopf algebras. This duality relates a braiding of thecategory to a quasitriangular structure of the algebra.The approach of this paper is to start with a semisimple tensor category C with extrastructure, together with a functor F : C → Vec, that we understand as naturally associatedto C . From this perspective, it follows from [59] that a necessary and sufficient conditionfor F to be upgraded to a weak quasi-tensor functor is that ρ → dim( F ( ρ )) be a weakdimension function, meaning that dim( F ( ρ ⊗ σ )) ≤ dim( F ( ρ ))dim( F ( σ )) for all irreducibleobjects ρ , σ . A weak quasi-tensor structure with the same dimension function is notunique, but passing to another affects the weak quasi-Hopf algebra by a twist deformation.We discuss two main istances, Wenzl functor for the fusion category C ( g , q, ℓ ) of quantumgroups at roots of unity, and Zhu’s functor for the fusion category of a vertex operatoralgebra.A second main result of our paper is the construction of w-Hopf algebras A W ( g , q, ℓ )associated to Wenzl’s functore W : C ( g , q, ℓ ) → Vec for all certain primitive roots of unity q with sufficiently large order. When q is a minimal root, we show that A W ( g , q, ℓ ) isunitary ribbon and its antipode is of a Kac type in a certain sense motivated by thetheory of compact quantum groups. This extends a result previously shown in [23] for thecase g = s l N with different methods. This twist connects A W (( g , q, ℓ ) to the Zhu algebra A Z ( V g k ) of the affine Lie algebra V g k for a suitable positive integer k and makes it into acoboundary weak quasi-Hopf algebra. In particular, Rep( V g k ) becomes a unitary modulartensor category in this way and we refer to the abstract for more details on our conjecturalequivalence with Huang-Lepowsky structure.An aspect making the C*-case of interest is that there are cases where Ω admits asquare root twist, that is a twist T such that Ω = T ∗ T , Ω − = T − ( T − ) ∗ . In this waythe Ω-involution of a unitary weak quasi-Hopf algebra can be twisted into one in theusual sense that is the ∗ -involution commutes with the coproduct. While this square rootconstruction always exists for Ω-involutive quasi-Hopf C ∗ -algebras with Ω positive, it isnot clear whether the same holds in the weak case. One of the main result of our paperis the construction of a square root in the weak C ∗ -case for the unitary structure of the w -Hopf algebra A W ( g , q, ℓ ) making also the braiding in a very simple ”exponential” form,that is a connection with the original Drinfeld-Kohno theorem. We wish to study possibleconnections with the braiding structure arising from loop group fusion categories in thework of A. Wassermann, Toledano-Laredo, Gui in later works.By general reasons due to amenability properties, a unitary w -Hopf algebra such that theinvolution commutes with the coproduct has necessarily integer dimensions, and thereforethe untwisted algebra while having trivial R -matrix and unitary structure, must have a 3-coboundary non-trivial associator. In particular, twisting our w -Hopf algebras A W ( g , q, ℓ )by a twist arising from the unitary structure as above, necessarily gives a unitary weak EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 7 quasi-Hopf algebras with a cohomologically trivial associator which is not a w -Hopf algebra.This may be identified with the structure of a unitary coboundary weak-quasi-Hopf algebraon the Zhu algebra associated to the corresponding affine VOA following Wenzl continuouspath recalled in the abstract, see also Sect. 19–25.Our work has been inspired and motivated by many excellent books and papers in theliterature. If we had to choose just a few, it is quite fare to say that we were inspiredby the approach to tannakian duality in the book by Neshveyev and Tuset, and the the-ory of compact quantum groups, that we suitably modify to treat our main examples.We were also inspired by the paper by Bakalov-Kirillov presentation of Drinfeld-Kohno-Kazhdan-Lusztig theorem end by Neshveyev and Tuset proof, based on Tannakian dualityand discrete algebras, and also by the work by Neshveyev-Yamashita on classification ofcompact quantum group. Furthermore, the book by EGNO also has been an importantreference to us.In the final version we would like to revise the introduction, add a section concerningpositivity of hermitian forms in pointed fusion categories, revise Sect. 17, expand thediscussion around Huang and Lepowsky tensor product theory, expand our application ofthe Drinfeld-Kohno theorem from the perspective of affine VOAs concerning Knizhnik-Zamolodgikov equations of CFT, and hopefully compare with some of the work of A.Wassermann and Gui, and a revise references list on this area.This is a very preliminary version of a paper written over a period of several years. Asthe paper is not complete, CP feels to take responsability on the actual content, especiallycorrectedness issues.2. Preliminaries on tensor categories and their functors
In this section we recall the the basic terminology concerning tensor categories andunitary tensor categories. Our main references are [37, 96] and [100] respectively. We alsogive the main definitions of certain functors between these categories. The most familiarnotion is that of tensor functor but we need suitable weak generalizations, known in theliterature as quasi-tensor functors and more importantly for us their weak versions, the weak quasi-tensor functors . We also introduce a new notion, that of weak tensor functor between tensor categories as a slight generalisation of notions already considered in theliterature. We shall describe a cohomological interpretation in the setting of weak quasiHopf algebras later on. Finally, we introduce a notion of unitarity for weak quasi-tensorfunctors between unitary tensor categories and discuss a unitarization procedure for generalweak quasi-tensor functors which will be fruitful later on.All categories in this paper will be essentially small, thus they will admit a small skeleton.The morphism space from an object ρ to σ is denoted by ( ρ, σ ). By a linear category C we mean a category whose morphism spaces are complex vector spaces and such thatcomposition is bilinear.The notion of semisimple category is central in this paper, we briefly recall the definitiondirecting our attention to linear categories, we refer the reader to Ch. 1 in [37] for details. S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI A linear additive category C is a linear category with a zero object 0, that is (0 ,
0) = 0,and direct sums, that is for any pair of objects ρ , σ ∈ C there is an object τ ∈ C andmorphisms S ∈ ( ρ, τ ), T ∈ ( σ, τ ), S ′ ∈ ( τ, ρ ), T ′ ∈ ( τ, σ ) such that S ′ S = 1, T ′ T = 1, SS ′ + T T ′ = 1. The object τ is defined up to isomorphism and denoted ρ ⊕ σ . A linearabelian category is a linear additive category with extra structure. The central additionalnotion is that of kernel and symmetrically of cokernel of a morphism. For a morphism A ∈ ( ρ, σ ) the kernel Ker( A ) is an object k and a morphism K ∈ ( k, ρ ) such that AK = 0,and universal with this property. Kernels and cokernels are assumed to exist for everymorphism, among other things. A subobject of an object ρ is an object σ together with amorphism S ∈ ( σ, ρ ) with Ker( S ) = 0. An object ρ is called simple , or irreducible , if ρ = 0and the only subobjects are 0 and ρ .It follows from Schur’s Lemma, see e.g. Lemma 1.5.2 in [37] and Prop. 5.4.5 in [24]that in a linear abelian category with finite dimensional morphism spaces, when ρ and σ are simple, ( ρ, σ ) is either the trivial vector space or it is formed by scalar multiples of aunique isomorphism, it follows that ( ρ, ρ ) = C
1. In our paper, all our categories will havefinite dimensional morphism spaces.A semisimple category is a linear abelian category such that every nonzero object is afinite direct sum of simple objects, the decomposition is unique up to isomorphism.A splitting idempotent, or a summand, of an object ρ is an object σ , an idempotent E ∈ ( ρ, ρ ) together with morphisms S ∈ ( σ, ρ ), S ′ ∈ ( ρ, σ ) such that S ′ S = 1, SS ′ = E . Inparticular, σ is a subobject of ρ . For example, a direct sum ρ ⊕ σ as previously defined has ρ and σ as summands defined by complementary idempotents. In a semisimple categoryevery idempotent splits, thus every subobject is a summand.The next notion is that of tensor category . We follow Sect. 1.2 in [96], and the notionof monoidal category of Ch. 2 in [37] except for we assume the linear structure. By a tensor category we mean a linear category C endowed with a tensor product operation ⊗ ,which is a bilinear bifunctor C × C → C , a distinguished tensor unit object ι and naturalisomorphisms α ρ,σ,τ : ( ρ ⊗ σ ) ⊗ τ → ρ ⊗ ( σ ⊗ τ ).The associativity morphisms α ρ,σ,τ satisfy the pentagon equation (( ρ ⊗ σ ) ⊗ τ ) ⊗ υ α (cid:15) (cid:15) α ⊗ / / ( ρ ⊗ ( σ ⊗ τ )) ⊗ υ α / / ρ ⊗ (( σ ⊗ τ ) ⊗ υ ) ⊗ α (cid:15) (cid:15) ( ρ ⊗ σ ) ⊗ ( τ ⊗ υ ) α / / ρ ⊗ ( σ ⊗ ( τ ⊗ υ )) (2.1)The tensor unit ι satisfies the unit axioms , that is the functors ρ → ρ ⊗ ι and ρ → ι ⊗ ρ are autoequivalences of C . By Sect. 2.9 in [37] one can identify ρ ⊗ ι and ι ⊗ ρ by a simplepassage which uses only the unit isomorphisms, in this way ι becomes strict, meaning that ι ⊗ ρ = ρ ⊗ ι = ρ for every object and 1 ι ⊗ T = T ⊗ ι = T for every morphism T . Tosimplify our discussion, we shall assume that ι is strict in our abstract results, and we shalltacitly use this passage in our applications where it is not natural to work with a strictunit, e.g. Sect. 17. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 9
We shall only deal with tensor categories for which the tensor unit satisfies ( ι, ι ) = C C , we fix aset Irr( C ) of simple objects in C such that every simple object of C is isomorphic to exactlyone element of Irr( C ). This can be done because we are assuming that C is essentiallysmall. A semisimple tensor category with finitely many inequivalent irreducible objectswill be called finite semisimple . If C is in addition rigid, it is a (complex) fusion category[38].Functors between linear categories are C -linear maps between morphism spaces. Definition 2.1.
A linear functor F : C → C ′ between linear categories is called a linearequivalence if there is a linear functor, called a quasi-inverse , G : C ′ → C such that FG and GF are naturally isomorphic to the identity functors of C ′ and C respectively. Remark 2.2.
It is well known (Theorem 1 in IV.4 of [85]) that a linear functor F : C → C ′ is a linear equivalence if and only if it is full and faithful (i.e. bijective between themorphism spaces) and essentially surjective (every object of C ′ is isomorphic to one inthe image of F .) We shall use these definitions interchangeably. When C and C ′ aresemisimple, this is equivalent to the property that { F ( ρ ) , ρ ∈ Irr( C ) } is a complete set ofpairwise non-isomorphic simple objects in C ′ .The following notion of weak quasi-tensor functor was introduced by H¨aring-Oldenburgin [59] in connection with the study of duality for weak quasi-Hopf algebras. Definition 2.3.
Let C and C ′ be tensor categories. A weak quasi-tensor functor is definedby a C -linear functor F : C → C ′ satisfying F ( ι ) = ι together with two morphisms F ρ,σ : F ( ρ ) ⊗ F ( σ ) → F ( ρ ⊗ σ ) and G ρ,σ : F ( ρ ⊗ σ ) → F ( ρ ) ⊗ F ( σ ) satisfying F ι,ρ = F ρ,ι = 1 F ( ρ ) , G ι,ρ = G ρ,ι = 1 F ( ρ ) , (2.2) F ρ,σ ◦ G ρ,σ = 1 F ( ρ ⊗ σ ) (2.3) F ρ ′ ,σ ′ ◦ F ( S ) ⊗ F ( T ) = F ( S ⊗ T ) ◦ F ρ,σ , F ( S ) ⊗ F ( T ) ◦ G ρ,σ = G ρ ′ ,σ ′ ◦ F ( S ⊗ T ) (2.4)for objects ρ , σ , ρ ′ , σ ′ ∈ C and morphisms S : ρ → ρ ′ , T : σ → σ ′ .Property (2.4) expresses naturality of F and G in ρ and σ , while the right inversecondition (2.3) implies that P ρ,σ = G ρ,σ ◦ F ρ,σ : F ( ρ ) ⊗ F ( σ ) → F ( ρ ) ⊗ F ( σ ) (2.5)is an idempotent satisfying F ρ,σ ◦ P ρ,σ = F ρ,σ , P ρ,σ G ρ,σ = G ρ,σ . If P ρ,σ = 1 F ( ρ ) ⊗ F ( σ ) for all ρ , σ (i.e. all F ρ,σ are isomorphisms), we recover the notion ofquasi-tensor functor of [34, 89]. Definition 2.4.
Let F , F ′ : C → C ′ be two weak quasi-tensor functors defined by( F ρ,σ , G ρ,σ ), ( F ′ ρ,σ , G ′ ρ,σ ), respectively. A natural transformation η : F → F ′ is called monoidal if η ι = 1 ι and if F ′ ρ,σ ◦ η ρ ⊗ η σ = η ρ ⊗ σ ◦ F ρ,σ , G ′ ρ,σ ◦ η ρ ⊗ σ = η ρ ⊗ η σ ◦ G ρ,σ . A weak quasi-tensor functor ignores the associativity structure of C and C ′ . The followingdefinition is motivated by the requirement of compatibility between the functor and theassociativity morphisms. Definition 2.5.
Let C and C ′ be tensor categories with associativity morphisms α and α ′ respectively. A weak tensor functor is a weak quasi-tensor functor F : C → C ′ for whichthe associated natural transformations F ρ,σ , G ρ,σ satisfy F ( α ρ,σ,τ ) = F ρ,σ ⊗ τ ◦ F ( ρ ) ⊗ F σ,τ ◦ α ′ F ( ρ ) , F ( σ ) , F ( τ ) ◦ G ρ,σ ⊗ F ( τ ) ◦ G ρ ⊗ σ,τ (2.6) F ( α − ρ,σ,τ ) = F ρ ⊗ σ,τ ◦ F ρ,σ ⊗ F ( τ ) ◦ α ′− F ( ρ ) , F ( σ ) , F ( τ ) ◦ F ( ρ ) ⊗ G σ,τ ◦ G ρ,σ ⊗ τ . (2.7)In the case that all F ρ,σ are isomorphisms then G ρ,σ = F − ρ,σ thus only one of the equations(2.6) and (2.7) suffices and we recover the notion of a tensor functor [37, 73, 96, 100]. Definition 2.6. A tensor equivalence between tensor categories C and C ′ is a tensor functor E : C → C ′ which is an equivalence of linear categories.It is known that a quasi-inverse G : C ′ → C may be chosen tensorial and the naturaltransformations 1 C ′ → FG , 1 C → GF monoidal see Remark 2.4.10 in [37]. In particular, G is a tensor equivalence as well.In general, we are making no assumption on compatibility of α ′ with the two subobjectsof ( F ( ρ ) ⊗ F ( σ )) ⊗ F ( τ ) and F ( ρ ) ⊗ ( F ( σ ) ⊗ F ( τ )) corresponding respectively to the rightinvertible maps F ρ ⊗ σ,τ ◦ F ρ,σ ⊗ F ( τ ) and F ρ,σ ⊗ τ ◦ F ( ρ ) ⊗ F σ,τ .A weak quasi-tensor functor monoidally isomorphic to a weak tensor functor is itselfweak tensor.The notion of weak (quasi) tensoriality for a functor applies to contravariant functors C → C ′ as well, but in this case the defining natural transformations are required to act as F ρ,σ : F ( ρ ) ⊗ F ( σ ) → F ( σ ⊗ ρ ), G ρ,σ : F ( σ ⊗ ρ ) → F ( ρ ) ⊗ F ( σ ) and the diagrams (2.6) and(2.7) have to be appropriately modified. Equivalently, such functors may be regarded ascovariant (quasi) tensor functors after replacing C ′ with the opposite category ( C ′ ) op thatis the category with same objects and morphisms, but opposed morphisms and reversedtensor products.We shall also consider categories with involutions and involution preserving functors.We shall follow [31] and [100]. These structures will not be needed until Sect. 9. Definition 2.7. A ∗ -category is a linear category C endowed with an antilinear, con-travariant, involutive functor ∗ : C → C acting trivially on objects. A tensor ∗ -category is atensor category equipped with the structure of a ∗ -category satisfying ( S ⊗ T ) ∗ = S ∗ ⊗ T ∗ for any pair of morphisms S , T ∈ C . The associativity morphisms are assumed unitary, α ∗ ρ,σ,τ = α − ρ,σ,τ . EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 11
Definition 2.8. A C*-category is a ∗ -category where morphism spaces are Banach spacessuch that the norm satisfies k S ◦ T k ≤ k S kk T k and k T ∗ ◦ T k = k T k for every pair ofmorphisms S , T and S ∗ S is positive (i.e. has positive spectrum) in the algebra ( ρ, ρ ) forevery morphism S ∈ ( ρ, σ ). Finally, a tensor C ∗ -category is a tensor ∗ -category which isalso a C*-category with respect to the given ∗ -involution.The positivity condition is equivalent to the existence of S ′ ∈ ( ρ, ρ ) such that S ∗ S = S ′∗ S ′ . It follows in particular that ( ρ, ρ ) is a C ∗ -algebra for any object ρ . In a C ∗ -category,two isomorphic objects ρ , σ are called unitarily isomorphic if there is a unitary U ∈ ( ρ, σ ),that is U ∗ U = 1, U U ∗ = 1. An orthogonal summand of ρ is a summand defined by aselfadjoint idempotent E ∈ ( ρ, ρ ) which is the range of an isometry (there is S ∈ ( σ, ρ )such that S ∗ S = 1 and SS ∗ = E ). An orthogonal direct sum ρ ⊕ σ is defined by isometries S ∈ ( ρ, ρ ⊕ σ ), S ∈ ( σ, ρ ⊕ σ ) such that S S ∗ + S S ∗ = 1.It follows from the positivity of T ∗ T that a left invertible morphism T ∈ ( σ, ρ ) admitspolar decomposition in C . Thus S = T ( T ∗ T ) − / ∈ ( σ, ρ ) is an isometry. In particular,two isomorphic objects ρ , σ are also unitarily isomorphic.It also follows that a summand or a direct sum is isomorphic to an orthogonal one. In-deed, by Prop. 4.6.2 in [13] every idempotent in a unital C ∗ -algebra is similar to a selfad-joint idempotent. Thus a summand σ of ρ up to isomorphism corresponds to a selfadjointidempotent in E ∈ ( ρ, ρ ), and it follows that polar decomposition of the correspondingmorphism S ∈ ( σ, ρ ) gives the needed isometry. Similarly, the defining complementaryidempotents of a direct sum ρ ⊕ σ may be assumed selfadjoint and it follows that thedirect sum is orthogonal.In particular, a semisimple C ∗ -category has orthogonal summands and direct sums. Itis also easy to see that the positivity condition of T ∗ T follows from the other properties ofa C ∗ -category and existence of orthogonal direct sums, cf. Ch. 2 in [100]. Definition 2.9. A ∗ -functor F : C → C ′ between ∗ -categories is a linear functor satisfying F ( T ∗ ) = F ( T ) ∗ for all morphisms T ∈ C . If C and C ′ are tensor ∗ -categories, a ∗ -functorendowed with a weak quasi tensor structure will be called a weak quasi tensor ∗ -functor .Let F be a weak quasi tensor ∗ -functor defined by F ρ,σ , G ρ,σ . Then the adjoint pair F ′ ρ,σ = G ∗ ρ,σ , G ′ ρ,σ = F ∗ ρ,σ defines another weak quasi tensor structure on F . Definition 2.10. A ∗ -equivalence between ∗ -categories C and C ′ is an equivalence compati-ble with the ∗ -structure, that is a ∗ -functor E : C → C ′ admitting a quasi-inverse E ′ : C ′ → C which is a ∗ -functor with natural unitary transformations η : 1 → EE ′ and η ′ : 1 → E ′ E . If C and C ′ are tensor ∗ -categories, E is a tensor ∗ -equivalence if E and E ′ are tensor ∗ -functors.We note the following C*-version of the characterisation of equivalences between cate-gories of Remark 2.2. Proposition 2.11.
Let F : C → C ′ be a ∗ -functor between C*–categories. Then F is a ∗ -equivalence if and only if it is a ∗ -functor which is an equivalence of linear categories. If C and C ′ are tensor C ∗ -categories then F is a tensor ∗ –equivalence if and only if it is a ∗ -functor and a tensor equivalence. Proof.
We start with the definition of a linear equivalence as a full, faithful and essentiallysurjective functor F , as in Remark 2.2. Theorem IV.4.1 [85] constructs a linear functor G : C ′ → C and invertible natural transformations η : 1 → FG and η ′ : 1 → GF . Weare thus left to show that we can always choose η and η ′ unitary and G a ∗ -functor. Tothis aim, it is not difficult to adapt the proof of that theorem to the needed frameworkas follows. The isomorphisms η c defined there, corresponding to our η , may be chosenunitary passing to polar decomposition available with the C*-structure of C ′ . This impliesthat the quasi-inverse equivalence constructed there and denoted T , in turn correspondingto G , satisfies that η : 1 → FG is a unitary natural transformation. This fact, togetherwith the fact that F is a faithful ∗ -functor, implies that G is linear and ∗ -preserving onmorphism spaces. If η ′′ : 1 → GF is any invertible natural transformation, one of which isfound in the same theorem, then the unitary part in the polar decomposition η ′ of η ′′ willbe a unitary natural transformation between the same functors thanks to the ∗ -preservingproperties of the involved functors. The last statement follows from the fact that when C and C ’ are tensor C ∗ -categories then we already know that we may construct a tensorialquasi-inverse G and then we apply the first part of the proof. (cid:3) Remark 2.12.
We note that a faithful and essentially surjective ∗ -functor between ∗ -categories F : C → C ′ does not necessarily admit a quasi-inverse ∗ -functor. An example isgiven by the immersion of the category Hilb of finite dimensional Hilbert spaces into thecategory Herm of finite dimensional Hermitian spaces. This category will be introducedand studied starting with Sect. 9. For the subclass of semisimple ∗ -categories we havethe following useful criterion analogous in analogy to Remark 2.2. Let Irr u ( C ) be a set ofpairwise unitarily inequivalent simple objects in C such that every other simple object isunitarily isomorphic to one element of Irr u ( C ). A faithful ∗ -functor between ∗ -categories F : C → C ′ is a ∗ -equivalence if and only if the set of objects F ( ρ ) with ρ ∈ Irr u ( C ) is acomplete set of pairwise unitarily inequivalent simple objects in C ′ .In the theory of C ∗ -tensor categories, or more generally of tensor ∗ -categories, we havethe following notion of unitarity for a tensor functor and a tensor equivalence, see [100]. Definition 2.13.
Let C and C ′ be tensor ∗ -categories. A unitary tensor functor ( F , F, G = F − ), is a tensor ∗ -functor such that F is unitary. A unitary tensor equivalence is a tensor ∗ -equivalence which is unitary as a tensor ∗ -functor and with a unitary quasi-inverse.Unitary tensor functors from C ∗ -tensor categories to Hilb arise as forgetful functors ofcompact quantum groups see e.g. [100]. As fusion categories do not in general admit tensorfunctors to Vec, but always admit weak quasi-tensor functors, we introduce a notion ofunitarity in the following more general setting.We next begin to discuss a problem that has relevence in how paper, that is how toassociate to a given weak quasi-tensor structure ( F, G ) another one that has in some sensea more trivial unitary structure. Historically, the first condition considered in the literatureis G = F ∗ and G unitary see e.g. [100], or more generally isometry [59]. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 13
Definition 2.14.
Let C and C ′ be tensor ∗ -categories. A unitary weak quasi tensor functor is a weak quasi tensor ∗ -functor F : C → C ′ defined by ( F, G ) such that F ∗ and G areisometries. A strongly unitary weak quasi tensor functor we have in addition F ∗ = G .For quasi-tensor ∗ -functors we recover the usual notion of unitarity F ∗ ρ,σ = F − ρ,σ . Remark 2.15.
The definition of unitarity may equivalently be formulated by the proper-ties F ∗ ρ,σ F ρ,σ = P ∗ ρ,σ P ρ,σ , G ρ,σ G ∗ ρ,σ = P ρ,σ P ∗ ρ,σ where P ρ,σ is the idempotent defined in (2.5).In general, if ( F, G ) is unitary then we may have two new strongly unitary structures(
F, F ∗ ) and ( G ∗ , G ) arising from ( F, G ). However, in the C ∗ -case all these structurescoincide. More precisely, we note the following simple result. Proposition 2.16.
Let C be a tensor ∗ -category, C ′ a tensor C ∗ -category and ( F, G ) aweak quasi-tensor structure for a ∗ -functor F : C → C ′ . Let ρ , σ ∈ C be a pair of objects. If F ∗ ρ,σ and G ρ,σ are isometries then F ∗ ρ,σ = G ρ,σ . In particular, any unitary weak quasi-tensorstructure is automatically strongly unitary.Proof. We have that F ρ,σ G ρ,σ = 1 F ( ρ ⊗ σ ) = G ∗ ρ,σ G ρ,σ = F ρ,σ F ∗ ρ,σ . It follows that G ∗ ρ,σ (1 F ( ρ ) ⊗ F ( σ ) − F ∗ ρ,σ F ρ,σ ) G ρ,σ = G ∗ ρ,σ G ρ,σ − ( F ρ,σ G ρ,σ ) ∗ ( F ρ,σ G ρ,σ ) = 0 . The C ∗ -property of C ′ implies (1 − F ∗ ρ,σ F ρ,σ ) G ρ,σ = 0 thus G ρ,σ = F ∗ ρ,σ . (cid:3) To construct unitary weak quasi-tensor structures from a given weak quasi-tensor struc-ture, structure it is natural to try with polar decomposition.We consider a weak quasi tensor ∗ -functor ( F , F, G ) : C → C ′ between C ∗ -tensor cat-egories and we describe a a unitarization of the weak quasitensor structure ( F, G ). Weset Ω ρ,σ := F ∗ ρ,σ ◦ F ρ,σ ∈ ( F ( ρ ) ⊗ F ( σ ) , F ( ρ ) ⊗ F ( σ )) . Note that Ω ρ,σ is partially invertible (in the sense of Def. 4.1) with partial inverseΩ − ρ,σ := G ρ,σ ◦ G ∗ ρ,σ ∈ ( F ( ρ ) ⊗ F ( σ ) , F ( ρ ) ⊗ F ( σ ))satisfying Ω − ρ,σ Ω ρ,σ = P ρ,σ and Ω ρ,σ Ω − ρ,σ = P ∗ ρ,σ . Since they are both positive, we may takethe respective square roots Ω / ρ,σ and (Ω − ρ,σ ) / .If we know that (Ω − ρ,σ ) / is a left inverse of Ω / ρ,σ , that is(Ω − ρ,σ ) / Ω / ρ,σ = P ρ,σ (2.8)then we shall just write Ω − / ρ,σ for (Ω − ρ,σ ) / . We have F = S ∗ ◦ Ω / , G = Ω − / ◦ T, (2.9)where S and T are isometries as G is a right inverse of F . Proposition 2.17.
Let F : C → C ′ be a weak quasi-tensor ∗ -functor between tensor C ∗ -categories defined by ( F, G ) such that (Ω − ρ,σ ) / Ω / ρ,σ = P ρ,σ (e.g. P = 1 ). Then a) the pair ( F ′ , G ′ ) , where F ′ = F Ω − / = S ∗ Ω / Ω − / , G ′ = Ω / G = Ω / Ω − / T, is a unitary weak quasi-tensor structure for F , and therefore strongly unitary, F ′ = G ′∗ , b) In particular, if ( F , F, G ) is quasi tensor then F ′ = S ∗ , G ′ = S = T is always welldefined and is a unitary quasi-tensor structure, c) if F is full and if ( F, G ) is a tensor structure then ( S ∗ , S ) is a unitary tensorstructure for F .Proof. a) It follows from ∗ -invariance of F that Ω ρ,σ is natural in ρ , σ , and from continuousfunctional calculus that Ω / ρ,σ and Ω − / ρ,σ are natural as well, hence the same holds for F ′ and G ′ . We have F ′ G ′ = F P G = 1, so ( F ′ , G ′ ) is a weak quasi tensor structure. Theassociated idempotent is given by P ′ := G ′ F ′ = Ω / P Ω − / = Ω / Ω − / . Furthermore F ′ F ′∗ = F (Ω − ) / (Ω − ) / F ∗ = F GG ∗ F ∗ = 1, G ′∗ G ′ = G ∗ Ω / Ω / G = G ∗ F ∗ F G = 1,thus ( F ′ , G ′ ) is unitary, and by Prop. 2.16 also strongly unitary. c) In this case F , G are invertible and G = F − , thus P = 1, S , T are unitary and S ∗ T = 1. d) Since( G ∗ ρ,σ ◦ G ρ,σ ) / is a positive invertible element in the C ∗ -algebra ( F ( ρ ⊗ σ ) , F ( ρ ⊗ σ )) and F is full, we may write ( G ∗ ρ,σ ◦ G ρ,σ ) / = F ( A ρ,σ ) with A ρ,σ ∈ ( ρ ⊗ σ, ρ ⊗ σ ) positive, and G ρ,σ = S ρ,σ ◦ F ( A ρ,σ ) with S unitary. It follows that F (1 ρ ⊗ A σ,τ ) is positive by ∗ -invarianceof F and also invertible by naturality of G . Furthermore,1 F ( ρ ) ⊗ G σ,τ ◦ G ρ,σ ⊗ τ = 1 F ( ρ ) ⊗ S σ,τ ◦ F (1 ρ ) ⊗ F ( A σ,τ ) ◦ G ρ,σ ⊗ τ =1 F ( ρ ) ⊗ S σ,τ ◦ G ρ,σ ⊗ τ ◦ F (1 ρ ⊗ A σ,τ ) = 1 F ( ρ ) ⊗ S σ,τ ◦ S ρ,σ ⊗ τ ◦ B ρ,σ,τ , where B ρ,σ,τ := F ( A ρ,σ ⊗ τ ) ◦ F (1 ρ ⊗ A σ,τ ) . A similar computation starting with the sameelement but relying now on naturality of S in place of G , see a), leads to conclude that F ( A ρ,σ ⊗ τ ) and F (1 ρ ⊗ A σ,τ ) commute, and this implies that B ρ,σ,τ is positive, besidesinvertible. In a similar way G ρ,σ ⊗ F ( τ ) ◦ G ρ ⊗ σ,τ = S ρ,σ ⊗ F ( τ ) ◦ S ρ ⊗ σ,τ ◦ C ρ,σ,τ for someother positive invertible morphism C ρ,σ,τ . Inserting these relations into the tensorialitydiagram 1 F ( ρ ) ⊗ G σ,τ ◦ G ρ,σ ⊗ τ ◦ F ( α ) = α ′ ◦ G ρ,σ ⊗ F ( τ ) ◦ G ρ ⊗ σ,τ gives another tensorialitydiagram satisfied by S in place of G by unitarity of the associativity morphisms anduniqueness of polar decomposition. (cid:3) Definition 2.18.
Let ( F , F, G ) : C → C ′ be a weak quasi-tensor ∗ -functor between tensor C ∗ -categories satisfying the left inverse property (2.8). Then the same functor F togetherwith the new unitary weak quasi-tensor structure ( F ′ , G ′ ) defined in part a) of Prop. 2.17will be called the unitarization of ( F , F, G ). Remark 2.19.
We would like to warn the reader that it is not clear to us whether (2.8)holds in our main late applications as in Sect. 23 and following. It follows that it is unclearwhether the polar decomposition construction of Prop. 2.17 can be used. We shall needto develop a modification of the unitarization construction for a functor in Sect. 23. Onthe other hand, the unitarization of a functor will be fruitful for us in case of full domains( P = 1), see Sect. 14, where we shall discuss uniqueness of unitary structures in tensorcategories. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 15
Part c) shows that in the important case of tensor ∗ -equivalence the unitarization givesa unitary tensor equivalence. We have the following consequence. Corollary 2.20.
Two tensor ∗ -equivalent tensor C ∗ -categories are also unitarily tensorequivalent. Remark 2.21.
Note that we do not have a statement about unitarization of a weaktensor ∗ -functor. On this subject we shall see that the notion of unitary weak tensor ∗ -functor is too strong for unitary fusion categories of interest for us. Specifically, a unitaryweak tensor ∗ -functor to the category of Hilbert spaces is automatically tensor for largeclasses of semisimple unitary tensor tensor categories and the category necessarily has aninteger-valued dimension function, we refer to Corollary 13.8 for details. It follows thatthe unitarization of a weak tensor ∗ -functor in general is only a unitary weak quasi-tensor ∗ -functor. In Sects. 24 we shall construct examples of weak tensor ∗ -functors associated tounitary fusion categories of quantum groups at roots of unity, and part a) of Prop. 2.17will turn out useful.3. Rigidity, ribbon category and weak tensor functors
In this brief section we recall the notion of rigidity, braided and ribbon tensor categoryand we show a simple result that weak tensor functors are always compatible with rigidity.
Definition 3.1.
Let C be a tensor category with associativity morphisms α ρ,σ,τ ∈ (( ρ ⊗ σ ) ⊗ τ, ρ ⊗ ( σ ⊗ τ )). An object ρ ∨ is a right dual of ρ if there are morphisms d ∈ ( ρ ∨ ⊗ ρ, ι )and b ∈ ( ι, ρ ⊗ ρ ∨ ) satisfying the right duality equations1 ρ ⊗ d ◦ α ρ,ρ ∨ ,ρ ◦ b ⊗ ρ = 1 ρ , (3.1) d ⊗ ρ ∨ ◦ α − ρ ∨ ,ρ,ρ ∨ ◦ ρ ∨ ⊗ b = 1 ρ ∨ . (3.2)A left dual object ∨ ρ is defined by morphisms b ′ ∈ ( ι, ∨ ρ ⊗ ρ ), d ′ ∈ ( ρ ⊗ ∨ ρ, ι ) satisfying theleft duality equations d ′ ⊗ ρ ◦ α − ρ, ∨ ρ,ρ ◦ ρ ⊗ b ′ = 1 ρ , (3.3)1 ∨ ρ ⊗ d ′ ◦ α ∨ ρ,ρ, ∨ ρ ◦ b ′ ⊗ ∨ ρ = 1 ∨ ρ . (3.4)A tensor category is called rigid if every object has left and right duals.The following facts are well known: another right dual ( ˜ ρ, ˜ b, ˜ d ) is isomorphic to ρ ∨ , theisomorphism is ξ := ˜ d ⊗ ρ ∨ ◦ ˜ ρ ⊗ b : ˜ ρ → ρ ∨ , (3.5)and similarly for left duals. If ρ and σ have right duals ρ ∨ and σ ∨ , then so does ρ ⊗ σ , and itis given by σ ∨ ⊗ ρ ∨ via the morphisms d ρ ⊗ σ = d σ ◦ σ ∨ ⊗ ( d ρ ⊗ σ ) ◦ α ∈ (( σ ∨ ⊗ ρ ∨ ) ⊗ ( ρ ⊗ σ ) , ι ), b ρ ⊗ σ = α ′ ◦ ρ ⊗ ( b σ ⊗ ρ ∨ ) ◦ b ρ ∈ ( ι, ( ρ ⊗ σ ) ⊗ ( σ ∨ ⊗ ρ ∨ )), where α and α ′ are suitableassociativity morphisms. Definition 3.2. A right duality is defined by the choice of a right dual ( ρ ∨ , b ρ , d ρ ) for eachobject ρ such that ι ∨ = ι with b ι = d ι = 1 ι . A left duality is defined in a similar way. Every right duality gives rise to a contravariant functor D : C → C acting as ρ → ρ ∨ , T ∈ ( ρ, σ ) → T ∨ := d σ ⊗ ρ ∨ ◦ σ ∨ ⊗ T ⊗ ρ ∨ ◦ σ ∨ ⊗ b ρ ∈ ( σ ∨ , ρ ∨ ) , (3.6)called the right duality functor , which turns out tensorial. A different right duality structure( ˜ ρ, ˜ b ρ , ˜ d ρ ) gives a corresponding duality functor ˜ D related to D via the isomorphisms ξ ρ :˜ ρ → ρ ∨ defined in (3.5), which is a natural monoidal isomorphism ξ : ˜ D → D .Right and left dualities naturally arise in representation categories of Hopf algebras andtheir generalisations, where canonical choices are induced by the antipode, we shall discussthis in detail in Sect. 5. A well-behaved choice of right and left dualities lead to thenotion of spherical category . In a spherical category a theory of categorical dimension canbe developed. By a theorem of Deligne [136], see also Sect. 20, when the category isbraided there is a correspondence between spherical structures and ribbon structures forthe braided symmetry. Definition 3.3.
Let C be a tensor category with right duality ( ρ ∨ , b ρ , d ρ ). A naturalisomorphism η ∈ (1 ,
1) of the identity functor of C is called compatible with duality if η ρ ∨ = ( η ρ ) ∨ . We recall the definition of braided symmetry and ribbon category.
Definition 3.4. A braided symmetry for C is a natural isomorphism c ( ρ, σ ) ∈ ( ρ ⊗ σ, σ ⊗ ρ )such that c ( ρ, ι ) = c ( ι, ρ ) = 1 ρ (3.7)and the following two hexagonal diagrams commute( ρ ⊗ σ ) ⊗ τ α −−−→ ρ ⊗ ( σ ⊗ τ ) c −−−→ ( σ ⊗ τ ) ⊗ ρ c ⊗ y y α ( σ ⊗ ρ ) ⊗ τ α −−−→ σ ⊗ ( ρ ⊗ τ ) ⊗ c −−−→ σ ⊗ ( τ ⊗ ρ ) (3.8)( ρ ⊗ σ ) ⊗ τ c −−−→ τ ⊗ ( ρ ⊗ σ ) α − −−−→ ( τ ⊗ ρ ) ⊗ σ α − x y c ⊗ ρ ⊗ ( σ ⊗ τ ) ⊗ c −−−→ ρ ⊗ ( τ ⊗ ρ ) α − −−−→ ( ρ ⊗ τ ) ⊗ σ (3.9)One may verify that the property of being compatible with duality for an isomorphism η ∈ (1 ,
1) does not depend on the choice of the right duality.
Definition 3.5.
Let C be a rigid tensor category with braided symmetry c . A ribbonstructure is a natural isomorphism v ∈ (1 ,
1) such that c ( σ, ρ ) ◦ c ( ρ, σ ) = v ρ ⊗ v σ ◦ v − ρ ⊗ σ andcompatible with some right duality.Unitary braided symmetries are central notions for this paper see e.g. Sect. 10, 21, 24,18, 17. In Sect. 20 we shall extend Deligne theorem to a class of symmetries more generalthan braided symmetries which play a central role in the study of unitary structures inthis paper in Sect. 21, 23, 24. Furthermore, ribbon structure and categorical dimensionare used in our applications, the classification result of s l N,ℓ -type categories in Sect. 18.
EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 17
Proposition 3.6.
Let ( F , F, G ) : C → C ′ be a weak tensor functor between tensor cate-gories. If ρ ∨ is a right dual of ρ defined by d ∈ ( ρ ∨ ⊗ ρ, ι ) and b ∈ ( ι, ρ ⊗ ρ ∨ ) then F ( ρ ∨ ) is a right dual of F ( ρ ) defined by d = F ( d ) ◦ F ρ ∨ ,ρ and b = G ρ,ρ ∨ ◦ F ( b ) , similarly for leftduals.Proof. We only show that d and b solve (3.1) for F ( ρ ). We have1 F ( ρ ) ⊗ d ◦ α ′ F ( ρ ) , F ( ρ ∨ ) , F ( ρ ) ◦ b ⊗ F ( ρ ) = F ρ,ι ◦ F ( ρ ) ⊗ d ◦ α ′ F ( ρ ) , F ( ρ ∨ ) , F ( ρ ) ◦ b ⊗ F ( ρ ) ◦ G ι,ρ = F ρ,ι ◦ F ( ρ ) ⊗ F ( d ) ◦ F ( ρ ) ⊗ F ρ ∨ ,ρ ◦ α ′ F ( ρ ) , F ( ρ ∨ ) , F ( ρ ) ◦ G ρ,ρ ∨ ⊗ F ( ρ ) ◦ F ( b ) ⊗ F ( ρ ) ◦ G ι,ρ = F (1 ρ ⊗ d ) ◦ F ρ,ρ ∨ ⊗ ρ ◦ F ( ρ ) ⊗ F ρ ∨ ,ρ ◦ α ′ F ( ρ ) , F ( ρ ∨ ) , F ( ρ ) ◦ G ρ,ρ ∨ ⊗ F ( ρ ) ◦ G ρ ⊗ ρ ∨ ,ρ ◦ F ( b ⊗ ρ ) = F (1 ρ ⊗ d ) ◦ F ( α ρ,ρ ∨ ,ρ ) ◦ F ( b ⊗ ρ ) = 1 F ( ρ ) . (cid:3) Corollary 3.7.
Let C be a rigid tensor category and F : C → Vec be a weak tensor functor.Then dim( F ( ρ )) = dim( F ( ρ ∨ )) = dim( F ( ∨ ρ )) for every object ρ . If a tensor category is rigid, left and right duals need not be isomorphic. It is easy tosee that this is the case if and only if ρ ≃ ρ ∨∨ and, following M¨uger, we call ρ ∨ a two-sideddual of ρ . We shall say that C has two-sided duals if every object has a two-sided dual.For example, duals are two-sided if C is a semisimple tensor category, see e.g. Prop. 2.1in [38], a tensor category with a coboundary, e.g. a braided symmetry, by Prop. 20.7, ora tensor ∗ -category [82]. In the last case, a solution d and b of the right duality equationsgives one of the left duality equations via ρ ∨ := ∨ ρ , b ′ = d ∗ and d ′ = b ∗ . This dual isalso called a conjugate of ρ and denoted ρ . The duality equations are written in terms of r := d ∗ and r := b , and referred to as the conjugate equations : r ∗ ⊗ ρ ◦ α − ρ,ρ,ρ ◦ ρ ⊗ r = 1 ρ ,r ∗ ⊗ ρ ◦ α − ρ,ρ,ρ ◦ ρ ⊗ r = 1 ρ . (3.10)Let C be a tensor C ∗ -category. The intrinsic dimension of ρ is defined as d ( ρ ) = inf k r k r k over all solutions of the conjugate equations for ρ [82]. Corollary 3.8.
Let C and C ′ be tensor C ∗ -categories and F : C → C ′ a weak tensor ∗ -functor defined by ( F, G ) . If ρ ∈ C has a conjugate then d ( F ( ρ )) ≤ k F ρ,ρ kk G ρ,ρ k d ( ρ ) .Proof. Let b , d solve the right duality equations for ρ and consider the associated solution b , d for F ( ρ ) as in Prop. 3.6, so r = d ∗ , r = b solves the conjugate equations forthe same object. We have r ∗ r ≤ k F ρ,ρ k F ( r ∗ r ) so k r k ≤ k F ρ,ρ kk r k by the C*-property.Similarly k r k ≤ k G ρ,ρ kk r k and the conclusion follows. (cid:3) In particular if F is a unitary weak tensor functor we have d ( F ( ρ )) ≤ k d ( ρ ) k , and if F is in turn unitary tensor we recover a well known upper bound in representation theory ofcompact quantum groups of the vector space dimension of a representation by the quantumdimension. More precisely, this case corresponds to C the representation category of thecompact quantum group, C ′ = Hilb and F the forgetful functor, see Cor. 2.2.20 in [100]. As already remarked before Def. 2.14, we shall see that by Prop. 3.8 together with theresults in Sect. 13 and more specifically Cor. 13.8, in C and C ′ are rigid C ∗ -tensor categoriesand C is amenable then every unitary weak tensor functor F : C → C ′ preserves the intrinsicdimensions. In particular, non-integrality of the intrinsic dimension is an obstruction tothe concurrence of both unitarity and weak tensoriality for a weak quasitensor structure( F , F, G ) to Hilb. In the non-weak case this result was shown in [82], see also Cor. 2.7.9 in[100] and references therein. Examples of non-unitary weak tensor structures or unitaryweak quasitensor structures arising from fusion categories associated to quantum groupsat roots of unity and conformal field theory will be discussed in Sect. 23, 18, 17.4. Weak quasi-Hopf algebras
In [34] Drinfeld introduced the notion of quasi-Hopf algebra as an extension of that ofHopf algebra to the case where the coproduct is not coassociative. Quasi-Hopf algebrasare more flexible than Hopf algebras in that they admit a so called twist operation.Quasi Hopf algebras play an important role in the proof of the Drinfeld-Kohno theoremon the connection between conformal field theory and quantum groups [34], see also [98].However, quasi-Hopf algebras are not sufficiently general to describe fusion categories fromCFT. This follows from Frobenius-Perron theorem, according to which a fusion category C admits a unique positive dimension function, it is the Frobenius-Perron dimension function, ρ ∈ Irr( C ) → FPdim( ρ ), see Sect. 5 in [37], see also Sect. 13, 18. This implies that C istensor equivalent to Rep( A ) for a quasi-Hopf algebra A if and only if FPdim takes valuesin N , in this case A is unique up to twist deformation. However the integrality conditionis not satisfied already for the fusion category associated the Ising model, which may berealised by an affine vertex operator algebra over s l at level 2 [87].In the early 90s Mack and Schomerus [87] suggested to give up the request that thecoproduct is unital. This leads to the notion of weak quasi Hopf algebra, that is the mainsubject of this section and plays a central role in this paper. As we shall see, Drinfeldnotion of twist deformation extends in a natural way to weak quasi-Hopf algebras. Definition 4.1.
Let B be an algebra, and consider the linear category with objects idem-potents of B and morphism spaces between two idempotents p , q ∈ B defined by( p, q ) := qBp = { T ∈ B : qT = T = T p } . Given an element T ∈ ( p, q ), we shall refer to D ( T ) := p and R ( T ) := q as the domain and range of T . We shall call T partially invertible if it is invertible as a morphism of thatcategory. In other words, if there is an element T − ∈ ( q, p ) satisfying T − T = p, T T − = q. (4.1)Clearly T − is unique in ( q, p ). We shall refer to T − as the partial inverse, or simply theinverse of T .In most of our applications, p is given. Assume that we have T and T − such that T − is a partial left inverse of T in the sense of the first equation (4.1), then we have a uniquerange q = T T − such that T is partially invertible. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 19
Definition 4.2. A weak quasi-bialgebra A is defined by the following data a) algebra : a complex, associative algebra A with unit I , b) coproduct : a possibly non-unital homomorphism ∆ : A → A ⊗ A c) counit : a homomorphism ε : A → C satisfying( ε ⊗ ◦ ∆ = 1 = (1 ⊗ ε ) ◦ ∆ , (4.2) d) associator : a partially invertible element Φ ∈ A ⊗ A ⊗ A with D (Φ) = ∆ ⊗ I )) , R (Φ) = 1 ⊗ ∆(∆( I )) , (4.3)Φ∆ ⊗ a )) = 1 ⊗ ∆(∆( a ))Φ , a ∈ A, (4.4)(1 ⊗ ⊗ ∆(Φ))(∆ ⊗ ⊗ I ⊗ Φ)(1 ⊗ ∆ ⊗ ⊗ I ) , (4.5)1 ⊗ ε ⊗ I ) . (4.6)The relations ε ⊗ ⊗ I ) = 1 ⊗ ⊗ ε (Φ) hold automatically, a result extendinga known result for quasi-bialgebras algebras. For example, the first follows from the factthat the domain and range of ε ⊗ ⊗ I ), and then as in the quasi-bialgebra case[34], one evaluates ε ⊗ ε ⊗ ⊗ Definition 4.3. A weak quasi-Hopf algebra is a weak quasi-bialgebra with an antipode : anantiautomorphism S of A together with elements α , β ∈ A for which S ( a (1) ) αa (2) = ε ( a ) α, a (1) βS ( a (2) ) = ε ( a ) β, a ∈ A (4.7) xβS ( y ) αz = I = S ( x ′ ) αy ′ βS ( z ′ ) , (4.8)where m : A ⊗ A → A is the multiplication map and we use the notation Φ = x ⊗ y ⊗ z ,Φ − = x ′ ⊗ y ′ ⊗ z ′ .If ∆ is unital, the definition of weak quasi-Hopf algebra reduces to that of quasi-Hopfalgebra introduced by Drinfeld in [34]. The following example provides the simplest familyof quasi-Hopf algebras. Example 4.4.
Let G be a finite group. The algebra Fun ω ( G ) of complex valued func-tions on G is a commutative quasi-bialgebra with coproduct ∆( f )( g, h ) = f ( gh ), counit ε ( f ) = f ( e ), associator given by a normalized 3-cocycle ω : G → T . If ω is triv-ial we recover the usual Hopf algebra Fun( G ). If ω is a 3-cocycle and ω F ( g, h, k ) = F ( h, k ) F ( g, hk ) ω ( g, h, k ) F − ( gh, k ) F − ( g, h ) is a cohomologous 3-cocycle via a normal-ized 2-cochain F then Fun ω F ( G ) = (Fun( G ) ω ) F . It follows that the twist isomorphismclass of Fun ω ( G ) is determined by the class of ω in H ( G, T ). An antipode is given by S ( f )( g ) = f ( g − ), α ( g ) = ω ( g, g − , g ) − , β ( g ) = 1. (Note that the 3-cocycle relation for ω yields the equality ω ( g, g − , g ) = ω ( g − , g, g − ) − , which is useful to verify the antipodeaxioms.) Definition 4.5.
An antipode (
S, α, β ) will be called strong if α = β = I . Remark 4.6.
An antiautomorphism S of A can be a strong antipode only if it satisfiesthe following compatibility conditions with the associator, xS ( y ) z = I, S ( x ′ ) y ′ S ( z ′ ) = I. (4.9)For example, when A is a bialgebra, that is Φ = I ⊗ I ⊗ I , then the above equationsobviously hold and the notion of a strong antipode reduces to the usual notion of antipodeof a Hopf algebra. More generally, in the weak case we shall see that equations (4.9) aresatisfied by the associator of a w -Hopf algebra, see Sect. 6. Definition 4.7.
Let A be a weak quasi-bialgebra with coproduct ∆ and counit ε .a) A twist is a pair of elements T, T − ∈ A ⊗ A such that T − is a partial left inverseof T , that is T − T = ∆( I ) and such that ε ⊗ T ) = 1 ⊗ ε ( T ) = I .b) A trivial twist of A is a twist of the form E = P ∆( I ) where P ∈ A ⊗ A is anidempotent, E − = ∆( I ) P , EE − = P .If P is a trivial twist then P = ∆ P ( I ). In particular, in the framework of quasi-bialgebrasthe only trivial twist is the identity, and this motivates our terminology. Trivial twists mayinformally be thought as the necessary adjustment between two weak bialgebra structuresthat that would be coinciding except for the value the coproducts take on the identity.Trivial twists will arise in the study of unitary structures in Sect. 7 and unitary ribbonstructures in Sect. 21, 23, 24. Proposition 4.8.
A twist T of a weak quasi bialgebra A gives rise to another weak quasi-bialgebra, denoted A T , with the same algebra structure and counit as A but coproduct andassociator given by ∆ T ( a ) = T ∆( a ) T − Φ T = I ⊗ T ⊗ ∆( T )Φ∆ ⊗ T − ) T − ⊗ I. (4.10) If A has antipode ( S, α, β ) , then A T has antipode ( S, α T , β T ) where α T = S ( f ′ ) αg ′ , β T = f βS ( g ) , (4.11) and T = f ⊗ g , T − = f ′ ⊗ g ′ .Proof. Verification of the axioms can be done as in the unital case, [73], with slight modi-fications due to non triviality of domain idempotents. (cid:3)
In the last part of the section we extend to weak quasi-Hopf algebras properties ofantipodes of quasi-Hopf algebras [34].
Proposition 4.9.
Let A be a weak quasi-Hopf algebra with antipode ( S, α, β ) . Then forevery invertible u ∈ A , the triple ( S, α, β ) defined by S ( a ) = uS ( a ) u − , (4.12) α = uα, β = βu − (4.13) is another antipode of A . Conversely, any antipode is of this form with u ∈ A uniquelydetermined by (4.12) and one of the equations in (4.13). EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 21
Proof.
From (4.5) it follows that(1 ⊗ ⊗ ∆(Φ))(∆ ⊗ ⊗ − ⊗ I ) =( I ⊗ Φ)(1 ⊗ ∆ ⊗ ⊗ ∆(∆( I ))) ⊗ I ) = ( I ⊗ Φ)(1 ⊗ ∆ ⊗ . We may extend the proof of the quasi-Hopf case, i.e. Prop. 1.1 of [34], to the weakcase. (cid:3)
Notice that u and u − can be derived from (4.13) if one of the antipodes is strong. Corollary 4.10.
Let A be a weak quasi-Hopf algebra and ( S, α, β ) an antipode. Then a) A admits a strong antipode if and only if α and β are invertible and β = α − . Inparticular a strong antipode is unique and given by ad( α − ) S . b) If A admits a strong antipode S then the same holds for a twisted algebra A T if andonly if m ◦ S ⊗ T − ) = ( m ◦ ⊗ S ( T )) − . (4.14) Proof.
The proof follows from (4.11) and Prop. 4.9. (cid:3)
By [34], p. 1424, when Φ = I , thus A is a bialgebra, and ( S, α, β ) is an antipode then β = α − , thus we may always assume that the antipode is strong. We shall see that thisproperty extends to any w -bialgebra with an antipode of a weak quasi-Hopf algebra, seeProp. 6.5. We illustrate these notions for the quasi-Hopf algebras defined in Example 4.4. Example 4.11.
It follows from Cor. 4.10 that A = Fun ω ( G ) has a strong antipode if andonly if ω ( g, g − , g ) = 1 for all g . For example, when G = Z N , each complex N -th rootof unity w induces the 3-cocycle ω w ( a, b, c ) = w γ ( a,b ) c , with γ ( a, b ) = ⌊ a + bN ⌋ − ⌊ aN ⌋ − ⌊ bN ⌋ ,where ⌊ λ ⌋ is the greatest integer not exceeding λ . Furthermore this association gives anisomorphism of the group of N -th roots of unity with H ( Z N , T ). If h is the naturalgenerator of Z N , ω ( h, h − , h ) = w . It follows that Fun ω w ( Z N ) ∈ H if and only if w = 1.Quite interestingly, elements of Fun ω ( G ) ∈ H ′ can be twisted to elements of H whichare not Hopf algebras, but this can happen only if a certain obstruction of the associatorvanishes. More in detail, F is a twist such that (Fun ω ( G )) F ∈ H if and only if β F = α − F which amounts to solve the equation F ( g − , g ) ω ( g, g − , g ) = F ( g, g − ) (4.15)When there are elements g ∈ G with g = e and such that ω ( g, g, g ) = 1 then clearly theequation has no solution. For example, for G = Z , ω − ( h, h, h ) = −
1. Note that this is ageneral property, λ g := ω ( g, g, g ) = ± g = e , and it is not difficult to see that theproperty that λ take the value − g is the only obstruction tosolve equation (4.15) for a normalized twist F . For example the obstruction vanishes if G has odd order. We shall come back to 3-cocycles on Z N in Sect. 18, cf. (18.1).Drinfeld showed that the antipode of a quasi-Hopf algebra satisfies a twisted antico-multiplicativity property with the coproduct which extends the usual (i.e. untwisted)anticomultiplicativity in the framework of Hopf algebras. We in turn extend this to weak quasi-Hopf algebras. Since our arguments are a straightforward generalisation of [34], weshall only briefly sketch the needed modifications. Set γ = V (( I ⊗ Φ − )(1 ⊗ ⊗ ∆(Φ))) , δ = V ′ ((∆ ⊗ ⊗ − ⊗ I )) (4.16)where V, V ′ : A ⊗ → A ⊗ are defined by V ( a ⊗ b ⊗ c ⊗ d ) = S ( b ) αc ⊗ S ( a ) αd and V ′ ( a ⊗ b ⊗ c ⊗ d ) = aβS ( d ) ⊗ bβS ( c ). Proposition 4.12.
Let A be a weak quasi-Hopf algebra. Then the new weak quasi-Hopfalgebra with same algebra structure and counit but coproduct S ⊗ S ◦ ∆ op ◦ S − and associator S ⊗ S ⊗ S (Φ ) is a twist of A by a unique partially invertible element f ∈ A ⊗ A suchthat γ = f · ∆( α ) , δ = ∆( β ) · f − . (4.17) Explicitly, D ( f ) = ∆( I ) , R ( f ) = S ⊗ S ◦ ∆ op ( I ) , f ∆( S ( a )) f − = S ⊗ S (∆ op ( a )) , S ⊗ S ⊗ S (Φ ) = Φ f . (4.18) We have f = S ⊗ S (∆ op ( p )) γ ∆( qβS ( r )) and f − = ∆( S ( p ) αq ) δS ⊗ S (∆ op ( r )) . In partic-ular, if the antipode is strong then f = γ , f − = δ .Proof. The proof of the first relation in (4.18) follows from the following two lemmas, inturn extending Lemmas 1 and 2 of [34] to weak quasi-Hopf algebras. More precisely, thanksto Lemma 4.13 we may apply lemma 4.14 to B = A ⊗ A , p = ∆( I ), q = S ⊗ S (∆ op ( I )), f = ∆, g = ∆ ◦ S , ρ = ∆( α ), σ = ∆( β ), g = S ⊗ S ◦ ∆ op , ρ = γ , σ = δ . We omit the proofof the second relation of (4.18). (cid:3) Lemma 4.13.
We have: a) γ = V ((Φ ⊗ I )(∆ ⊗ ⊗ − ))) , δ = V ′ ((1 ⊗ ⊗ ∆(Φ − ))( I ⊗ Φ)) , (4.19)b) for a ∈ A , ( S ⊗ S (∆ op ( a (1) ))) γ ∆( a (2) ) = ε ( a ) γ ∆( a (1) ) δ ( S ⊗ S (∆ op ( a (2) ))) = ε ( a ) δ (4.20)c) ∆( x ) δ ( S ⊗ S (∆ op ( y ))) γ ∆( z ) = ∆( I ) = (4.21)∆( I )( S ⊗ S (∆ op ( p ))) γ ∆( q ) δ ( S ⊗ S (∆ op ( r ))) (4.22) Proof. a) By the cocycle property (4.5) we can write γ = V ( I ⊗ (∆ ⊗ I )))(1 ⊗ ∆ ⊗ ⊗ I )(∆ ⊗ ⊗ − ))) . By the defining antipode property (4.7) we have, for T ∈ A ⊗ , V ( a ⊗ ∆( b ) ⊗ c · T ) = ε ( b ) V ( a ⊗ I ⊗ I ⊗ c · T ) = V (1 ⊗ ⊗ ε ⊗ a ⊗ b ⊗ c ) T ) . It suffices to choose a ⊗ b ⊗ c = I ⊗ ∆( I )Φ and T = (Φ ⊗ I )(∆ ⊗ ⊗ − )) . The identityinvolving δ can be proved in a similar way. The proof of b) and c) is a straightforwardgeneralisation of the case of quasi-Hopf algebras. We refrain from giving details, and werefer the interested reader to [34]. (cid:3) EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 23
Lemma 4.14.
Let B be a algebra, p an idempotent in B , f : A → B a homomorphismand g : A → B an anti-homomorphism with f ( I ) = g ( I ) = p , and ρ, σ ∈ pBp such that: g ( a (1) ) ρf ( a (2) ) = ε ( a ) ρ, f ( a (1) ) σg ( a (2) ) = ε ( a ) σ (4.23) where a ∈ A . Moreover, f ( x ) σg ( y ) ρf ( z ) = p, g ( p ) ρf ( q ) σg ( r ) = p (4.24) In addition, we have an idempotent q ∈ B , ρ, σ ∈ qBp and an anti-homomorphism g : A → B with g ( I ) = q also satisfying (4.23) - (4.24) (in (4.24) q replaces p ). Then thereexists a unique partially invertible element F ∈ B with D ( F ) = p , R ( F ) = q , such that F ρ = ρ, σF = σ (4.25) g ( a ) = F g ( a ) F − . (4.26) We have F = g ( p ) ρf ( q ) σg ( r ) , F − = X i g ( p ) ρf ( q ) σ g ( r ) . (4.27) Proof.
We first show uniqueness. Let F be partially invertible with the stated domain andrange and satisfying (4.25). Inserting the explicit form of p and q given in (4.24) in theequalities F = F p and F − = qF − , respectively, and taking into account the mentionedrelations (4.25), gives formulas (4.27).We apply the map W : A ⊗ → B , W ( b ⊗ c ⊗ d ) = g ( b ) ρf ( c ) σg ( d ), respectively to(∆ ⊗ a )))Φ − and Φ − (1 ⊗ ∆(∆( a ))) and obtain, if F is defined as in (4.27), F g ( a ) = g ( a ) F . Similarly, applying the map X : A ⊗ → B , X ( b ⊗ c ⊗ d ⊗ e ) = g ( b ) ρf ( c ) σg ( d ) ρf ( e ),to the equality: (1 ⊗ ⊗ ∆(Φ))(∆ ⊗ ⊗ − ⊗ I ) == ( I ⊗ Φ)(1 ⊗ ∆ ⊗ ⊗ ∆(∆( I )) ⊗ I )gives F ρ = ρ . The relations F F − = q , F − F = p follow again from (4.25). (cid:3) We next show that a strictly coassociative coproduct with trivial associator in the of aweak case, quasi-Hopf algebra is not compatible with non-unitality of the coproduct.
Proposition 4.15.
Let A be a weak quasi-Hopf algebra with coassociative coproduct andassociator Φ = ∆ ⊗ ◦ ∆( I ) = Φ − . Then A is a Hopf algebra.Proof. It is easy to see that Φ is an associator and that the elements α and β definingan antipode are invertible, hence A admits a strong antipode, say S . We are left toshow that ∆( I ) = I ⊗ I . The element γ defined by relation (4.16) turns out to be I thanks to coassociativity of ∆. Hence S satisfies the untwisted anticomultiplicative relation∆ ◦ S = S ⊗ S ◦ ∆ op by the previous proposition. We use the notation ∆( x ) = x ⊗ x and∆( I ) = a ⊗ b and compute∆( I ) = ∆( I ) ε ( a ) b ⊗ I = ∆( ε ( a ) I ) b ⊗ I = ∆( a S ( a )) b ⊗ I =∆( aS ( b )) b ⊗ I = a S ( b , ) b ⊗ a S ( b , ) = a S ( b , ) b , ⊗ a S ( b ) = a ε ( b ) ⊗ a S ( b ) = a ⊗ a S ( b ε ( b )) = a ⊗ a S ( b ) = a ⊗ b S ( b ) = a ⊗ ε ( b ) I = aε ( b ) ⊗ I = I ⊗ I. (cid:3) In conclusion of the section we introduce a class of most interest in this paper, thosefor which the underlying algebra is isomorphic to a direct sum of full matrix algebras.Although we are mostly interested in finite dimensional algebras, in the following definitionwe allow infinite dimensionality. The direct sum of full matrix algebras A = M r M n r ( C ) , is the algebra with elements of the form ( a r ) with a r ∈ M n r , and only finitely many ofthem are nonzero. The identity of M n r is a minimal central projection of A and will bedenoted by e r . Similarly, the direct product M ( A ) = Y r M n r ( C )is the algebra of elements ( a r ) of the same form but with no further restriction on theentries. There is no distinction between A and M ( A ) precisely when the index set is finite,which amounts to say that A is unital. Definition 4.16.
An algebra A is called discrete if it is isomorphic to a direct sum offull matrix algebras. A discrete weak quasi bialgebra (Hopf algebra) is a discrete algebraendowed with coproduct, counit and associator where the axioms of a weak quasi bialgebraare modified as follows. A coproduct ∆ : A → M ( A ⊗ A ) is assumed to take values in M ( A ⊗ A ) = Q r,s M n r ⊗ M n s . For fixed integers r , s , the sum P j ∆( e j ) e r ⊗ e r is welldefined as only finitely many entries are nonzero. Then the coproduct ∆ extends to amap M ( A ) → M ( A ⊗ A ) via the formula ∆( a ) e r ⊗ e s = P j ∆( a j ) e r ⊗ e s for a = ( a j ),and the extension is a homomorphism. In particular, ∆( I ) is a well defined idempotent of M ( A ⊗ A ). Similarly, ∆ ⊗ ⊗ ∆ extend to M ( A ⊗ A ). The associator Φ, counit ε (and the antipode ( S, α, β ) in the Hopf case) are defined as in the unital case, except thatΦ, α , β may lie in the corresponding multiplier algebras.Most of the results of this section hold for discrete weak quasi bialgebras (Hopf algebras).In Sect. 8 we shall introduce involutive and C*-versions. As we shall see in later sections,such a class is useful to study semisimple tensor categories. We also note that Van Daeledeveloped a theory for the multiplier Hopf algebras , a class of algebras more general thanthe discrete Hopf algebras [125]. An analogous generalization from the theory of weakquasi-Hopf algebras goes beyond the aim of this paper.5. Tannaka-Krein duality and integral weak dimension functions (wdf)
The problem of constructing weak quasi-Hopf algebras from an abstract fusion categorywas introduced in [87, 116] and developed in [59]. Their motivation was that the framework
EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 25 of quasi-Hopf algebras is an important notion for conformal field theory but too restric-tive for many related fusion categories as they may not admit integral valued dimensionfunctions. Their central idea consists in a weakened notion of a dimension function takingintegral values whose existence can easily be proven for all fusion categories and still allowsTannakian recostruction theorems. In this section we review and expand these results farbeyond fusion categories.In the first part of this section we describe how weak quasi-Hopf algebras lead to rigidtensor categories. We then discuss Tannaka-Krein duality results for semisimple rigidtensor categories. We shall then see that every fusion category may be described by a weakquasi-Hopf algebra associated to an integral weak dimension function on the Grothendieckring of the category. Moreover, we shall extend this result far beyond the class of fusioncategories.Our description originates from the work in [59] and will be fruitful later on, for differentpurposes. For example, the weak quasi-Hopf algebra representation provided by an integralweak dimension function provides a cohomological insight into the category that will befurther investigated in the paper. Moreover, weak dimension functions will play a centralrole in our study of unitary structures in fusion categories of affine vertex operator algebras.Furthermore, we shall describe examples of algebras naturally associated to certain fusioncategories for which the integral dimensions arising from their representations satisfy theweak dimension property, see Sect. 24 and 17.Let A be a complex unital algebra. By a representation of A we mean a unital left actionof A , ρ : A → L ( V ) on a finite dimensional complex vector space V . It is customary topass to the language of (left) A -modules, dropping reference to ρ . We shall conform to thisnotation when no confusion arises. The representation category Rep( A ) is the categorywith objects representations of A and morphisms between two objects the subspace ( ρ, ρ ′ )of L ( V ρ , V ρ ′ ) consisting of all A -linear maps. The forgetful functor is the functor F : Rep( A ) → Vecassociating a representation with its vector space, and acting trivially on morphisms.If A admits the structure of a weak quasi-bialgebra ( ε, ∆ , Φ) then the counit ε is a 1-dimensional representation. We may form the tensor product representation ρ ⊗ ρ ′ whichis the representation acting on the subspace V ρ ⊗ ρ ′ := ∆( I ) V ρ ⊗ V ρ ′ of the tensor product vector space V ρ ⊗ V ρ ′ with left action induced by the coproduct: ρ ⊗ ρ ′ := ρ ⊗ ρ ′ ◦ ∆ . Given two morphisms S ∈ ( ρ, σ ), S ′ ∈ ( ρ ′ , σ ′ ), the tensor product map S ⊗ S ′ ∈ L ( V ρ ⊗ V ρ ′ , V σ ⊗ V σ ′ ) commutes with the action of ∆( I ), thus takes V ρ ⊗ ρ ′ to V σ ⊗ σ ′ . The restriction S ⊗ T to V ρ ⊗ ρ ′ is a morphism in ( ρ ⊗ ρ ′ , σ ⊗ σ ′ ). Given representations ρ , σ , τ , ( ρ ⊗ σ ) ⊗ τ and ρ ⊗ ( σ ⊗ τ ) act respectively on the ranges of ∆ ⊗ ◦ ∆( I ) and 1 ⊗ ∆ ◦ ∆( I ). The restriction ofthe action of Φ to the space of ( ρ ⊗ σ ) ⊗ τ is an isomorphism α ρ,σ,τ : ( ρ ⊗ σ ) ⊗ τ → ρ ⊗ ( σ ⊗ τ ).In this way Rep( A ) becomes a tensor category with unit object the counit of A . Proposition 5.1.
The forgetful functor F : Rep( A ) → Vec of a weak quasi-bialgebra A isweak quasi-tensor with F ρ,σ = ∆( I ) and G ρ,σ the inclusion map. We give a categorical interpretation of the notion of twist of a weak quasi-Hopf algebra,extending properties known for quasi-Hopf algebras. Let A be a unital discrete algebraendowed with two weak quasi-bialgebra structures ( A, ε, ∆ , Φ), (
A, ε, ∆ ′ , Φ ′ ). We maycorrespondingly form two tensor categories Rep( A ), Rep ′ ( A ) and the functor E : Rep( A ) → Rep ′ ( A ) acting identically on objects and morphisms. This functor fixes the tensor units,it is full, faithful on morphisms and essentially surjective, and hence E is an equivalenceof linear categories. Furthermore, the two forgetful functors F : Rep( A ) → Vec, F ′ :Rep ′ ( A ) → Vec satisfy the property that F ′ E = F just as linear functors. We would liketo make E into an equivalence of tensor categories. Proposition 5.2.
Let the discrete unital algebra A be endowed with two weak quasi-bialgebra structures A = ( A, ε, ∆ , Φ) and A ′ = ( A, ε, ∆ ′ , Φ ′ ) . Then there is a bijectivecorrespondence between tensor structures on the identity linear equivalence E : Rep( A ) → Rep ′ ( A ) and twists F ∈ M ( A ⊗ A ) such that A ′ = A F as weak quasi-bialgebras. Given F ,the tensor structure E ρ,σ : E ρ ⊗ E σ → E ρ ⊗ σ is given by the action of F − .Proof. The proof is a straightforward extension of the case of quasi-bialgebras, for which werefer the reader to Prop. 2.1 in [98]. We briefly comment on how to construct the twist fromthe tensor structure. Given a tensor structure E ρ,σ on E : Rep( A ) → : Rep ′ ( A ) we considerthe unique elements F − , F ∈ M ( A ⊗ A ) having components E ρ,σ , and E ρ,σ − respectively inthe representation ρ ⊗ σ of A ⊗ A . Then ρ ⊗ σ ( F − F ) = E ρ,σ ◦ E − ρ,σ = 1 F ( ρ ⊗ σ ) = ρ ⊗ σ (∆( I )),hence F − F = ∆( I ). The relation ε ⊗ F ) = I = 1 ⊗ ε ( F ) can be checked in a similarway, hence F is a twist. The relations ∆ ′ = ∆ F and Φ ′ = Φ F correspond respectively tothe intertwining relations E ρ,σ ∈ ( E ρ ⊗ E σ , E ρ ⊗ σ ) and tensoriality property. (cid:3) Extending the terminology of [100] to non-coassociative Hopf algebras, a twist V ∈ A ⊗ A is called invariant if ∆ V = ∆ and Φ V = Φ. For example, if v ∈ A is central invertible then∆( v ) v − ⊗ v − is an invariant twist. By the previous proposition, invariant twists inducetensor autoequivalence structures on the identity functor Rep( A ) → Rep( A ) and they areall of this form in the discrete case.More generally, if A is discrete, given ( A, ε, ∆ , Φ) and (
A, ε, ∆ ′ , Φ ′ ), the weak quasi-tensor structures on E : Rep( A ) → Rep ′ ( A ) correspond to the twists F ∈ A ⊗ A such that∆ ′ = ∆ F . Given such a structure, the composite functor F ′ E becomes a weak quasi-tensorwith the composed structure. Since F = F ′ E as functors, this also induces a new weakquasi-tensor structure on F . Of course, this is given by the action of F − , with F thetwist corresponding to E , so the induced structure on F determines that of E . Thus theconstruction of a tensor structure on E can be regarded as that of a weak quasi-tensorstructure of the forgetful functor F : Rep( A ) → Vec defined by a twist F ∈ A ⊗ A solving( A, ε, ∆ ′ , Φ ′ ) = A F .Two weak quasi-tensor structures on F are monoidally isomorphic if and only if thecorresponding twists F and F are related by an invertible u ∈ M ( A ) such that F = EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 27 u ⊗ uF ∆( u − ). This corresponds to say that A F and A F are isomorphic as weak quasi-bialgebras. For example, the weak quasi-tensor structures on F monoidally isomorphic tothe original one correspond to twists of the form u ⊗ u ∆( u − ), where u ∈ A is an invertibleelement. These twists are called 2-coboundaries. The monoidal isomorphism η ρ acts as ρ ( u ) on V ρ .Rigidity in Rep( A ) is described similarly to quasi-Hopf algebras. Definition 5.3.
Let ρ be a representation of a weak quasi-Hopf algebra. The contragra-dient representations ρ c and c ρ are the the representations of A acting on the dual space V ′ ρ respectively as h ρ c ( a ) φ, ξ i = h φ, ρ ( S ( a )) ξ i , h c ρ ( a ) φ, ξ i = h φ, ρ ( S − ( a )) ξ i . Proposition 5.4. If A is a weak quasi-Hopf algebra the category Rep( A ) is rigid. Rightand left duals of an object ρ are respectively given by ρ ∨ = ρ c , ∨ ρ = c ρ. Solutions of the right and left duality equations are respectively given by d ρ ( φ ⊗ ξ ) = φ ( αξ ) b ρ = X i βe i ⊗ e i , and b ′ ρ = X i e i ⊗ S − ( β ) e i , d ′ ρ ( ξ ⊗ φ ) = φ ( S − ( α ) ξ ) where ( e i ) and ( e i ) is a dual pair of bases. Thus Rep( A ) is rigid and by the above proposition, an antipode of A induces right andleft duality structures, ( b ρ , d ρ ) and ( b ′ ρ , d ′ ρ ), respectively, and consequently a (say, right)duality functor c : ρ → ρ c acting as transposition of αT β on a morphism T . By Prop.4.12 the collection of operators f σ,ρ := Σ σ c ⊗ ρ c ( S − ⊗ S − ( f )) is an invertible naturaltransformation σ c ⊗ ρ c → ( ρ ⊗ σ ) c making c into a contravariant tensor functor. Wecompute the natural transformation associated to c . We canonically identify the doubledual space V ′′ ρ of a representation with V ρ , so ρ cc identifies with ρ ◦ S . Reading (4.18)as an intertwining relation f : ∆ → S ⊗ S ◦ ∆ op ◦ S − , it implies that S ⊗ S ( f − ) : S ⊗ S ◦ ∆ op ◦ S − → S ⊗ S ◦ ∆ ◦ S − , hence we can form the composite which intertwines S ⊗ S ( f − ) f : ∆ → S ⊗ S ◦ ∆ ◦ S − . This implies that ρ ⊗ σ ( f − S ⊗ S ( f )) can be regarded as an intertwiner ρ cc ⊗ σ cc → ( ρ ⊗ σ ) cc ,and this is the natural transformation of c .Note that left and right duals of the same object of Rep( A ) are equivalent whenever S is an inner automorphism of A and a converse holds if A is discrete, that is S is inducedby an invertible in M ( A ). For example if A is not assumed discrete, S is inner whenever A has an Ω-involution in the sense of the Sect. 8 commuting with S , by Cor. 8.17, or forthe class w-Hopf algebras introduced in Sect. 6 with a quasitriangular structure, by Prop.7.7. If S is inner, any invertible x ∈ A such that S ( a ) = xax − induces an invertible naturaltransformation η : 1 → c , where η ρ is defined by the action of ρ ( x ), but to construct apivotal structure we need a monoidal natural transformation. Definition 5.5. A pivotal weak quasi-Hopf algebra is a pair ( A, ω ) with A a weak quasi-Hopf algebra and ω ∈ A an invertible element, called the pivot , such that S ( a ) = ωaω − for all a ∈ A and f − S ⊗ S ( f ) = ∆( ω ) ω − ⊗ ω − .The pivot is not unique but determined up to multiplication by an invertible centralelement z satisfying ∆( z ) = z ⊗ z . In Sect. 7 we shall see that if A is a ribbon weak quasi-Hopf algebra, then there is a canonically associated ω such that η becomes a monoidal.Note that since the identity functor is tensorial, we may use this property to derive tenso-riality of c more easily for such class of algebras. Indeed, a quasi-tensor functor which ismonoidally isomorphic to a tensor functor must be tensorial as well. This endows Rep( A )with the structure of a pivotal tensor category. But more is true: Rep( A ) becomes aspherical category in the sense of [8], a result extending to the weak case, results knownfor ribbon Hopf algebras. Thus, there is a well-behaved theory of dimension in Rep( A ),see Sect. 13.The following Tannakian reconstruction results are due to [59] and extend to the weakcase an earlier result of Majid for discrete quasi-Hopf algebras [89]. For a review for discreteHopf algebras, see [94]. The starting point is an abstract semisimple category equippedwith a fibre functor F : C → Vec. We let Nat ( F ) denote the discrete algebra of naturaltransformations of F to itself with finite support. Theorem 5.6.
Let C be a semisimple category and F : C → Vec a faithful functor. Then (a) A = Nat ( F ) is a discrete algebra and there is a linear equivalence E : C → Rep( A ) which, after composition with the forgetful functor F A : Rep( A ) → Vec , is isomor-phic to F . Up to isomorphism, A is determined by the last property among discretealgebras. (b) If C is tensorial and F has a weak quasi-tensor structure then A is a weak quasi-bialgebra, E is a tensor equivalence, the isomorphism F A E ≃ F is monoidal and A is determined among discrete weak quasi-bialgebras.Let ( C , F ) satisfy the same assumptions as in (b) . (c) If C is braided then A is a quasitriangular weak quasi-bialgebra and E is braided. (d) If C is rigid and dim( F ( ρ )) = dim( F ( ρ ∨ )) then a solution of the right duality equa-tions induces an antipode on A making it into a weak quasi-Hopf algebra. (e) If C satisfies (d) and is ribbon then A is a ribbon weak quasi-Hopf algebra.Proof. We briefly discuss a few aspects that we shall need. (a) A natural transformation η ∈ Nat ( F ) = A is determined by the values it takes on a complete set of simple objects { ρ i } i , and this gives an algebra isomorphism of A ≃ L i L ( V i ), with V i = F ( ρ i ), so A isdiscrete. (b) As before, α ρ,σ,τ : ( ρ ⊗ σ ) ⊗ τ → ρ ⊗ ( σ ⊗ τ ) denote the associativity morphismsof C and F ρ,σ and G ρ,σ the natural transformations defining the quasi-tensor structure of F . Counit, coproduct, and associator of A are respectively defined as follows. We identify EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 29 A ⊗ A with natural transformations on two variables ζ ρ,σ : F ( ρ ) ⊗ F ( σ ) → F ( ρ ) ⊗ F ( σ ),and similarly for A ⊗ . We set: ε ( η ) = η ι , ∆( η ) ρ,σ = G ρ,σ ◦ η ρ ⊗ σ ◦ F ρ,σ , (5.1)Φ ρ,σ,τ = 1 F ( ρ ) ⊗ G σ,τ ◦ G ρ,σ ⊗ τ ◦ F ( α ρ,σ,τ ) ◦ F ρ ⊗ σ,τ ◦ F ρ,σ ⊗ F ( τ ) . It follows thatΦ − ρ,σ,τ = G ρ,σ ⊗ F ( τ ) ◦ G ρ ⊗ σ,τ ◦ F ( α − ρ,σ,τ ) ◦ F ρ,σ ⊗ τ ◦ F ( ρ ) ⊗ F σ,τ . The axioms can be checked with routine computations. In comparison with the quasi-tensor setting where the natural transformations are invertible, the relations F ρ,σ ◦ G ρ,σ =1 F ( ρ ⊗ σ ) is used here to show partial invertibility of Φ. The tensor equivalence E is F regarded as a functor with values in Rep( A ) and tensor structure obtained by restrictingthat of F . (c), (e) The notion of braided or ribbon tensor category is recalled in Sect. 3,Definitions 3.4 and 20.9 respectively. Quasitriangular and ribbon structures for weak quasi-bialgebras are given in Sect. 7, Definition 7.1 and 7.5. If c ( ρ, σ ) is a braided symmetryin C , and Σ( V, W ) is the permutation symmetry of Vec, then the element R ∈ M ( A ⊗ A )defined by Σ( F ( ρ ) , F ( σ )) ◦ R ρ,σ = G σ,ρ ◦ F ( c ( ρ, σ )) ◦ F ρ,σ makes A quasitriangular. When C has a ribbon structure v ρ then A has a ribbon structure defined by the ribbon element v ∈ M ( A ), where v is the natural transformation F ( v ρ ). (d) A weak quasi-Hopf algebraantipode ( S, α, β ) is constructed as follows. For ρ ∈ Irr( C ), we fix linear isomorphismsfrom the dual vector spaces U ρ : F ( ρ ) ′ → F ( ρ ∨ ), and extend U to a natural transformationfrom the functor ρ → F ( ρ ) ′ to the functor ρ → F ( ρ ∨ ). We set S ( η ) ρ = U tρ η tρ ∨ U tρ − , where L t : W ′ → V ′ is the transposed of the linear map L : V → W , and α , β are determined by F ( d ρ ) ◦ F ρ ∨ ,ρ ◦ U ρ ⊗ f ⊗ ξ ) = f ( α ρ ξ ), 1 ⊗ U − ρ ◦ G ρ,ρ ∨ ◦ F ( b ρ ) = P i β ρ e i ⊗ e i , for ρ ∈ Irr( C ), f ∈ F ( ρ ) ′ , ξ ∈ F ( ρ ), e i ∈ F ( ρ ) a linear basis and e i ∈ F ( ρ ) ′ the dual basis. We refer toLemma 12 in [59] or to Prop. 2.5 in [98] for the verification of the antipode axioms. (cid:3) Remark 5.7. a) By semisimplicity of C , faithfulness of F is equivalent to requiring that F ( ρ ) = 0 for all simple objects ρ . In particular, F is always faithful on the morphismspaces ( ρ, σ ) where both ρ and σ are = 0. b) The requirement of dimension equality in(d) is automatic in the case where C has finitely many inequivalent simple objects, (i.e.is a fusion category), see [98] for a discussion and references, and also where F is a weaktensor functor, by Cor. 3.7. c) When we start with a given semisimple weak quasi-Hopfalgebra A then Tannakian reconstruction of Theorem 5.6 applied to the forgetful functor F : Rep( A ) → Vec with the natural weak quasi-tensor structure provides a discrete weakquasi-Hopf structure on Nat ( F ) which corresponds to the original structure of A under thenatural inclusion of A with Nat ( F ). Note that the construction of an antipode of Nat ( F )as in the proof of Theorem 5.6 depends on the choice of a right duality ( ρ ∨ , b ρ , d ρ ) of Rep( A )and the natural transformation U . In particular, by Prop. 5.4 a given antipode ( S, α, β ) of A corresponds to the antipode of Nat ( F ) defined by the canonical right duality associatedto ρ ∨ = ρ c as in Prop. 5.4 and to the identity natural transformation U (note that this isan admissible choice as the functor ρ → F ( ρ ∨ ) coincides with ρ → F ( ρ ) ′ ). d) In general, the algebras Nat ( F ) and Nat( F ) of general natural transformations of F to itself may havedifferent representation categories, see [54]. However, regarding Nat( F ) = M (Nat ( F )) asa topological algebra with the strict topology defined by Nat ( F ) the category of nondegen-erate representations of Nat ( F ) coincides with the full subcategory of strictly continuousrepresentations of Nat( F ). We shall touch on again the relevance of the Tannakian alge-bra Nat( F ) as a topological algebra for the forgetful functor associated to U q ( g ) for theconstruction of the R -matrix, see Sects. 21, 23, 24.We next introduce the notion positive weak dimension function. Definition 5.8.
Let C be a semisimple tensor category. A positive weak dimension function is a positively valued function D defined on a complete set Irr( C ) of irreducible objectsand satisfying D ( ι ) = 1, and X τ ∈ Irr( C ) D ( τ )dim( τ, ρ ⊗ σ ) ≤ D ( ρ ) D ( σ ) . (5.2)When C is rigid a weak dimension function satisfying D ( ρ ) = D ( ρ ∨ ) = D ( ∨ ρ ), for all ρ , iscalled symmetric .If the inequality is always an equality we recover the notion of positive dimension func-tion. We tacitly extend a weak dimension function to all the objects of C via additivityand isomorphism invariance, and (5.2) reads as D ( ρ ⊗ σ ) ≤ D ( ρ ) D ( σ )for every pair of objects ρ and σ . A weak dimension function D for C may be regarded asa (unital, additive, and submultiplicative) function on the Grothendieck ring Gr( C ), andIrr( C ) as a Z -basis.For a large part of this paper, we shall consider weak dimension functions taking pos-itive integral values. Furthermore, when the categories have duals, we shall also assumethe symmetry condition. However, in Sect. 13 and 18 we shall also consider dimensionfunctions for a different purpose, which may not be positive or integral, but the contextshould lead to no confusion.If A is a weak quasi bialgebra and F : Rep( A ) → Vec is the forgetful functor of A then D ( ρ ) = dim( F ( ρ )) is an integral weak dimension function. It follows that a semisimple(rigid) tensor category C equivalent to the representation category of a weak quasi-bialgebra(quasi Hopf algebra) admits an integral (symmetric) weak dimension function. The fol-lowing result shows that under suitable conditions existence of an integral weak dimensionon C function is also a sufficient to represent C in this way. Theorem 5.9.
Let C be a semisimple linear category. (a) The assignment F → D , D ( ρ ) := dim( F ( ρ )) , is a bijective correspondence betweenfaithful functors F : C → Vec up to natural isomorphism and functions D : Irr( C ) → N . (b) If C is tensorial then the functor F admits a weak quasi-tensor structure if and onlyif D is an integral weak dimension function. Furthermore, quasi-tensor structurescorrespond to genuine dimension functions. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 31 (c)
The weak quasi bialgebra structures on A = Nat ( F ) associated to the various weakquasi-tensor structures on F of dimension D as in Theorem 5.6 are pairwise twistisomorphic.Proof. (a) Obviously naturally isomorphic functors are associated to the same function D : Irr( C ) → N . Conversely, given D , choosing, for ρ ∈ Irr( C ), a vector space F ( ρ )with dim( F ( ρ )) = D ( ρ ) gives rise to a faithful functor F : C → Vec, determined up tonatural isomorphism. (b) If F : C → Vec admits a (weak) quasi-tensor structure then D ( ρ ) := dim( F ( ρ )) is a (weak) dimension function. For the converse, since by assumption,dim( F ( ρ ) ⊗ F ( σ )) ≥ dim( F ( ρ ⊗ σ )) for all ρ , σ ∈ Irr( C ), we may pick epimorphisms F ρ,σ : F ( ρ ) ⊗ F ( σ ) → F ( ρ ⊗ σ ) and monomorphisms G ρ,σ : F ( ρ ⊗ σ ) → F ( ρ ) ⊗ F ( σ )satisfying F ρ,σ ◦ G ρ,σ = 1 and acting identically if either ρ or σ is the tensor unit. Weextend these maps to all the objects µ , ν using complete reducibility: choose α iρ ∈ ( ρ, µ ), β iρ ∈ ( µ, ρ ) with β jρ α iρ = δ i,j ρ , P i,ρ α iρ β iρ = 1 µ , and similarly for γ jσ ∈ ( σ, ν ), δ iσ ∈ ( ν, σ ).Set F µ,ν = P F ( α iρ ⊗ γ jσ ) ◦ F ρ,σ ◦ F ( β iρ ) ⊗ F ( δ jσ ). It is easy to see that naturality holds, thatis F µ ′ ,ν ′ ◦ F ( S ) ⊗ F ( T ) = F ( S ⊗ T ) ◦ F µ,ν . Naturality also shows that F µ,ν is independent ofthe choice of the morphisms involved in the decompositions. We similarly obtain a naturaltransformation G µ,ν and it is easy to see that F µ,ν ◦ G µ,ν = 1. We thus have a weak quasi-tensor structure, which is quasi-tensor if D is a dimension function. (c) If ( F, G ), ( F ′ , G ′ )define two weak quasi-tensor structures on F then we know from Theorem 5.6 and its proofthat the coproduct associated to the latter is defined by ∆ ′ ( η ) ρ,σ = G ′ ρ,σ ◦ η ρ ⊗ σ ◦ F ′ ρ,σ , andsimilarly for ∆. We may then write ∆ ′ ( η ) = G ′ F ∆( η ) GF ′ since F G = 1. Setting T = G ′ F and T − = GF ′ we see that these natural transformations may be regarded as elements of A ⊗ A and that T − T = GF = ∆( I ), T T − = G ′ F ′ = ∆ ′ ( I ). A similar computation showsthat the corresponding associators are related by the corresponding twist relation. (cid:3) It follows from Remark 14.4 and Theorem 5.9 that any finite semisimple (fusion) cat-egory is tensor equivalent to that of a weak quasi bialgebra (Hopf algebra), and a tensorequivalence corresponds to a twist isomorphism between two associated such algebras.
Corollary 5.10.
Let C and C ′ be semisimple tensor categories endowed with integral weakdimension functions D and D ′ respectively compatible with a linear equivalence E : C → C ′ .Then E admits the structure of a tensor equivalence if and only if the corresponding weakquasi-bialgebras are isomorphic up to twist.Proof. If the categories are tensor equivalent then we apply Th. 5.9 and Th. 5.6. Con-versely, let F : C → Vec and F ′ : C ′ → Vec be weak quasi-tensor functors of dimensions D and D ′ and associated weak quasi-bialgebras A and A ′ respectively. Then F ′ E and F : C → Vec have the same dimension D , so they are isomorphic by Th. 5.9 (a). It fol-lows that F ′ E admits a weak quasi-tensor structure with weak quasi-bialgebra isomorphicto A , thus there is a tensor equivalence E : C → Rep( A ) and a monoidal isomorphism F ′ E ≃ F A E with F A : Rep( A ) → Vec the forgetful functor. On the other hand, wesimilarly have a monoidal isomorphism of F ′ ≃ F A ′ E with E : C ′ → Rep( A ′ ) a tensorequivalence and F A ′ : Rep( A ′ ) → Vec the forgtful functor. Since A is isomorphic to atwist of A ′ , there is a tensor equivalence E : Rep( A ′ ) → Rep( A ) and an isomorphism F A E ≃ F A ′ by Prop. 5.2. We have an isomorphism of functors F A E ≃ F A E E E andsince E admits the structure of a tensor equivalence, the same holds for E E E . Let E ′ and E ′ be quasi-inverse tensor equivalences of E and E respectively. Then E ′ E ′ E E E isa tensor equivalence naturally isomorphic to E as a linear equivalence, thus E admits thestructure of a tensor equivalence. (cid:3) In Sect. 17 we shall use weak quasi-Hopf algebras associated to tensor equivalent fusioncategories to gain insight into the study of unitarizability of fusion categories and this willfind fruitful applications to CFT. We formulate a simple criterion that will eventually beuseful to construct ribbon tensor equivalences, see Sect. 18.If a weak quasi bialgebra A ′ is obtained from another such bialgebra A by replacing theassociator of the latter with a new one but leaving the rest of the structure unchanged,then Rep( A ) and Rep( A ′ ) have isomorphic Grothendieck rings. The following proposition,inspired by a similar statement in [101] for Hopf algebras, shows that at an abstract levelan isomorphism of Grothendieck rings of fusion categories can always be visualized in thisway. Proposition 5.11.
Let C and C ′ be semisimple tensor categories and let f : Gr( C ) → Gr( C ′ ) be an isomorphism between their Grothendieck rings. Let ( A, ∆ , Φ ′ ) be a weak quasibialgebra corresponding to an integral weak dimension function D ′ on C ′ . Then there is anassociator Φ for A defining a new weak quasi bialgebra ( A, ∆ , Φ) which corresponds to C with respect to D = D ′ f . In particular, if C ′ is a finite semisimple tensor category then C is tensor equivalent to one with the same category and tensor product structure as C ′ butpossibly different associativity morphisms.Proof. Consider a complete set Irr( C ′ ) of irreducible objects of C ′ . Let F ′ : C ′ → Vec bea weak quasi-tensor functor corresponding to D ′ and defining ( A, ∆ , Φ ′ ). By Theorem5.9 a weak quasi-tensor structure on F ′ is determined by the choice, for ρ , σ ∈ Irr( C ′ ),of (normalized) epimorphisms F ρ,σ : F ′ ( ρ ) ⊗ F ′ ( σ ) → F ′ ( ρ ⊗ σ ) and monomorphisms G ρ,σ : F ′ ( ρ ⊗ σ ) → F ′ ( ρ ) ⊗ F ′ ( σ ) satisfying F ρ,σ ◦ G ρ,σ = 1. These maps are in turn specifiedby the choice of linear maps maps G τ,iρ,σ : F ′ ( τ ) → F ′ ( ρ ) ⊗ F ′ ( σ ), F τ,jρ,σ : F ′ ( ρ ) ⊗ F ′ ( σ ) → F ′ ( τ ) for τ ∈ Irr( C ′ ), via P τ,i G τ,iρ,σ F ′ ( T τi ) =: G ρ,σ and P τ,i F ′ ( S τi ) F τ,iρ,σ =: F ρ,σ , where S τi ∈ ( τ, ρ ⊗ σ ), T τi ∈ ( ρ ⊗ σ, τ ) satisfy T τj S τi = δ i,j , P τ,i S τi T τi = 1, in turn subject to F τ,iρ,σ G υ,jρ,σ = δ τ,υ δ i,j . Writing A = Nat ( F ′ ), the coproduct formula of A given in (5.1) canbe written as ∆( η ) ρ,σ = P τ,i G τ,iρ,σ η τ F τ,iρ,σ by naturality of η .Note that we may establish a bijective correspondence ρ ∈ Irr( C ) → ρ ′ ∈ Irr( C ′ ) andlinear isomorphisms ( τ, ρ ⊗ σ ) → ( τ ′ , ρ ′ ⊗ σ ′ ). We then set F ( ρ ) := F ′ ( ρ ′ ), extend F to a faithful functor F : C → Vec, and consider the weak quasi-tensor structure of F defined by the same maps F τ,iρ,σ , G τ,iρ,σ under the correspondence ρ → ρ ′ . It follows that thecorresponding weak quasi bialgebras may be chosen with the same algebra and coproductstructures. (cid:3) Example 5.12.
Let G be a finite group. Consider the category C = Vec G of finitedimensional G -graded vector spaces with tensor product defined by convolution and trivialassociativity morphisms. The representation ring is Z G . The constant function D = 1 is EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 33 a dimension function, giving rise to the commutative bialgebra C ( G ) of complex functions f on G with usual coproduct ∆( f )( g, h ) = f ( gh ). Prop. 5.11 reduces to the knownclassification of tensor categories with this representation ring. Indeed, in this special caseit shows that any such category is tensor equivalent to some Vec ωG , obtained from Vec G witha new associativity morphism given by a normalised C × -valued 3-cocycle ω . It correspondsto the quasi-bialgebra C ω ( G ) coinciding with C ( G ) except for the associator, which is givenby ω . Since Vec ωG is a pointed fusion category, D = 1 is the only dimension function on Z G .Thus C ω ( G ) is, up to twist, the only quasi-bialgebra that can be associated to Vec ωG . Twistisomorphism corresponds to cohomologous cocycles. It follows that the fusion categoriesVec ωG are parameterised by H ( G, C × ). It also follows that Vec ωG admits a faithful tensorfunctor to Vec if and only if ω is cohomologically trivial.For example, the category Vec ω Z , with ω the non trivial element of H ( Z , C × ), arisesfrom the representation theory of the affine vertex operator algebra associated to s l atlevel 1, a topic that will be discussed in more detail in Sections 18, 17. We shall come backto this in more detail and generality later on. We shall see that this category also admitsa weak tensor functor to Vec with weak dimension function D ( ρ ) = 2, and ρ the uniquenon trivial irreducible object, cf. Example 15.1.The following result will be useful to construct a tensor structure on a given linearequivalence between semisimple tensor categories. Proposition 5.13.
Let C and C ′ be semisimple tensor categories, G : C → C ′ a tensorequivalence and F : C → C ′ a linear equivalence. If F and G induce the same isomorphismbetween the corresponding Grothendieck rings then F can be made into a tensor equivalence.Proof. By assumption, for every simple object ρ ∈ C , F ( ρ ) and G ( ρ ) are equivalent simpleobjects in C ′ , and any simple object of C ′ is equivalent to one of them. It follows that F and G are related by an invertible natural transformation η , and therefore F may be endowedwith a unique weak quasi-tensor structure making η monoidal. It also follows that this isa tensor structure for F since so is the quasi-tensor structure of G . (cid:3) w-Hopf algebras Hopf algebras are characterised among quasi Hopf algebras by the property of havingtrivial associator [34]. This characterization gives insight into the cohomological interpre-tation of quasi-Hopf algebras, in that it leads to the notion of a 3-coboundary associator.In this section we develop a weak analogue of the notion of Hopf algebra among weakquasi Hopf algebras. The corresponding special subclass will be termed w-Hopf algebras.We shall see that there is no strictly coassociative weak example, and we shall discussexamples later on.
Definition 6.1.
Let A be a weak quasi bialgebra with associator Φ and coproduct ∆. Weshall call Φ a 3-coboundary associator if there is a twist F ∈ A ⊗ A such thatΦ = 1 ⊗ ∆( F − ) I ⊗ F − F ⊗ I ∆ ⊗ F ) , (6.1) Φ − = ∆ ⊗ F − ) F − ⊗ II ⊗ F ⊗ ∆( F ) . (6.2)If A is a quasi bialgebra and F is an invertible twist then only one equation sufficesamong (6.1) and (6.2), and Def. 6.1 reduces to the corresponding notion of a 3-coboundaryassociator. We next introduce w-Hopf algebras.Let A be an algebra with a coproduct ∆ and a counit ε . To shorten some formulas, weset: P = ∆( I ) ,P = ∆ ⊗ P ) , Q = 1 ⊗ ∆( P ) ,P = ∆ ⊗ ⊗ P ) , Q = 1 ⊗ ⊗ ∆( Q )Assume that the coproduct is associative up to the following the following intertwiningrelations. For a ∈ A , Q ∆ ⊗ ◦ ∆( a ) = 1 ⊗ ∆ ◦ ∆( a ) P , (6.3) P ⊗ ∆ ◦ ∆( a ) = ∆ ⊗ ◦ ∆( a ) Q . (6.4) Proposition 6.2.
The element
Φ := Q P satisfies Def. 4.2 d), with partial inverse Φ − = P Q if and only if P Q P = P , Q P Q = Q , (6.5) Q ⊗ ∆ ⊗ I ⊗ P P ⊗ I ) P = Q ∆ ⊗ ∆( P ) P . (6.6) Proof.
Relations (6.5) correspond obviously to (4.3), and (6.3) to (4.4). We explicit thecocycle condition (4.5). We have I ⊗ P = 1 ⊗ ∆ ⊗ I ⊗ P ) and I ⊗ P Q = Q , andsimilarly Q ⊗ IP = P . This implies, taking into account (6.3) and (6.4), I ⊗ Φ1 ⊗ ∆ ⊗ ⊗ I = I ⊗ Q P ⊗ ∆ ⊗ Q P ) Q P ⊗ I = I ⊗ Q ⊗ ∆ ⊗ Q P ) P ⊗ I = I ⊗ Q ⊗ ∆ ⊗ Q )1 ⊗ ∆ ⊗ P ) P ⊗ I = I ⊗ Q ⊗ ∆ ⊗ ⊗ ∆( P )1 ⊗ ∆ ⊗ ⊗ P )) P ⊗ I = Q I ⊗ P Q ⊗ IP = Q ⊗ ∆ ⊗ I ⊗ P P ⊗ I ) P . On the other hand,1 ⊗ ⊗ ∆(Φ)∆ ⊗ ⊗ ⊗ ⊗ ∆( Q P )∆ ⊗ ⊗ Q P ) = Q ∆ ⊗ ∆( P ) P . Finally, the normalisation condition relation (4.6) is an immediate consequence of thecounit axioms (4.2). (cid:3)
Remark 6.3.
The cocycle relation (6.6) can alternatively be written as Q ⊗ ∆ ⊗ Q P ) P = Q ∆ ⊗ ∆( P ) P . Indeed, the computations in the last proof show that Q ⊗ ∆ ⊗ I ⊗ P ) = I ⊗ Q ⊗ ∆ ⊗ Q ) , (and a similar identity involving P and P ) hence multiplying on the left by Q , this termalso equals Q ⊗ ∆ ⊗ Q ). EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 35
Definition 6.4.
An algebra A with coproduct ∆ and counit ε for which the projections P , P j , Q j , j = 3 ,
4, satisfy the requirements of the previous proposition is a weak quasi-bialgebra with associator Φ = Q P and will be called a be called a w-bialgebra . Proposition 6.5.
If a w-bialgebra A admits an antipode ( S, α, β ) in the sense of weakquasi-Hopf algebras then α , β are invertible and β = α − . Hence ad( α − ) S is the uniquestrong antipode of A .Proof. A computation shows that if (4.7) holds for (
S, α, β ) where S is an antiautomor-phism of A , then equations (4.8) for the associator Φ = Q P reduce to βα = I and αβ = I . The last statement follows from Prop. 4.10 a) (cid:3) Definition 6.6.
A w-bialgebra with a (unique) strong antipode, will be called a w-Hopfalgebra . Remark 6.7.
The first examples of weak quasi-Hopf algebras appeared in the physicsliterature, in the work by Mack and Schomerus [87], who were motivated by the need ofconstructing a quantum analogue of a global gauge group for certain models of algebraicquantum field theories in low dimensions. They started with a nonsemisimple categoryof representations of U q ( s l ) at roots of unity and indicated how to construct a such analgebra [86, 87]. In a previous work [23], Mack-Schomerus construction was studied indetail in the more general case of U q ( s l N ), and it was shown that these are indeed w-Hopfalgebras in the sense of this section.We next state, without proof, a few simple properties of w-Hopf algebras (and in factalready of weak quasi-Hopf algebras) useful to construct new examples from given ones. Proposition 6.8.
Let A be a w-Hopf algebra. (a) (tensor products) If B is another w-Hopf algebra then the natural weak quasi-Hopfstructure on the tensor product algebra A ⊗ B is a w-Hopf algebra structure. (b) (subalgebras) let C be a unital subalgebra of A which is invariant under coproductand antipode. Then C is a w-Hopf algebra with the restricted structure and thereis a natural inclusion of rigid tensor categories Rep( A ) → Rep( B ) . (c) (quotients) If D is a w -Hopf algebra related to A via an algebra epimorphism A → D compatible with coproduct and antipode then there is an inclusion Rep( C ) → Rep( A ) as a full rigid tensor subcategory. Proposition 6.9.
Let A and B be w-Hopf algebras, and let α : A → B an algebra isomor-phism which intertwines the corresponding coproducts and antipodes. Then α is automati-cally an isomorphism of weak quasi-Hopf algebras. Semisimple bialgebras are described via Tannaka-Krein duality by semisimple tensorcategories endowed with a tensor functor to Vec. This characterization extends to w -bialgebras, and is based on the simple observation that they have a weak tensor furgetfulfunctor. Theorem 6.10.
Let C be a semisimple (rigid) tensor category with finite dimensionalmorphism spaces and F : C → Vec a faithful weak quasi-tensor functor (taking an object and a dual to spaces with the same dimension). Then A = Nat ( F ) is a w-bialgebra ( w -Hopf algebra) if and only if F is a weak tensor functor.Proof. Let A = Nat ( F ) be a w-bialgebra. The forgetful functor of A is weak tensor andthis implies that the same holds for F since it is monoidally isomorphic to the compositionof a tensor equivalence with the forgetful functor. Conversely, if F is weak tensor then theassociator Φ of A and its inverse Φ − are derived from (2.6) and (2.7), and a computationshows that Φ = 1 ⊗ ∆(∆( I ))∆ ⊗ I )), Φ − = ∆ ⊗ I ))1 ⊗ ∆(∆( I )), that is A is aw-bialgebra. For the last assertion note that the equality requirement on the dimensionsof an object and a dual are automatically satisfied in our case, thanks to Cor. 3.7. HenceTheorem 5.6 guaranties that A has an antipode. (cid:3) It follows that the constructions of Prop. 6.8 have a description in terms of pairs ofabstract tensor categories endowed with a weak tensor functor. In particular, the followingwill turn out useful to construct new w -Hopf algebras from given examples, see Sect. 24. Corollary 6.11.
Let C be a fusion category endowed with a weak tensor functor to Vec .Under Tannaka-Krein correspondence, full fusion subcategories D ⊂ C endowed with therestricted functor correspond to quotient w -Hopf algebras of A = Nat ( F ) . The class of w-Hopf is not invariant under general twists, but we next see that it is sounder a suitable subclass of twists, that play the role of 2-cocycles in our framework.
Definition 6.12.
Let A be a w-bialgebra. A twist F ∈ A ⊗ A is called a 2 -cocycle of A ifit satisfies the following equations,1 ⊗ ∆( F − ) I ⊗ F − F ⊗ I ∆ ⊗ F ) = Q P , (6.7)∆ ⊗ F − ) F − ⊗ II ⊗ F ⊗ ∆( F ) = P Q . (6.8)Note that P and P F := ∆ F ⊗ F F − ) are respectively domain and range for F ⊗ I ∆ ⊗ F ), and the partial inverse of this element is ∆ ⊗ F − ) F − ⊗
1, and similarly for I ⊗ F ⊗ ∆( F ). The 2-cocycle equations can equivalently be written in the following form Q F F ⊗ I ∆ ⊗ F ) = I ⊗ F ⊗ ∆( F ) P ,P F I ⊗ F ⊗ ∆( F ) = F ⊗ I ∆ ⊗ F ) Q , with Q F := 1 ⊗ ∆ F ( F F − ), as well as in a form which emphasises a categorical feature,Φ F F ⊗ I ∆ ⊗ F ) = I ⊗ F ⊗ ∆( F )Φ , Φ − F I ⊗ F ⊗ ∆( F ) = F ⊗ I ∆ ⊗ F )Φ − . This last form also shows that the notion of a 2-cocycle has an extension to weak quasi-Hopf algebras which in turn extends the corresponding notion for quasi-Hopf algebras, see,e.g., [73].
Proposition 6.13.
Let A be a weak quasi-bialgebra with coproduct ∆ and associator Φ ,and let F ∈ A ⊗ A be a twist. Then A F is a w-bialgebra if and only if Φ is the coboundaryassociator defined by F as in (6.1), (6.2). In particular, if A is a w-bialgebra, A F is aw-bialgebra as well if and only if F is a -cocycle. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 37
Proof.
We already know that A F is a weak quasi-bialgebra with coproduct ∆ F ( a ) = F ∆( a ) F − and associator Φ F = I ⊗ F ⊗ ∆( F )Φ∆ ⊗ F − ) F − ⊗ I . We have Φ − F = F ⊗ I ∆ ⊗ F )Φ − ⊗ ∆( F − ) I ⊗ F − . Hence for A F to be a w-bialgebra it suffices that theassociator and its inverse satisfy Φ F = Q F P F , Φ − F = P F Q F . A simple computation showsthat these equations are equivalent to the equations in the statement. If in particular A isa w-bialgebra as well, these equations reduce to the 2-cocycle equations (6.7), (6.8). (cid:3) Remark 6.14. If A is a quasi-Hopf algebra, equations (6.7) and (6.8) are precisely thecohomological equations which characterise a cohomologically trivial associator. Quite in-terestingly, these equations are meaningful for weak quasi-Hopf algebras with the weakcounterparts of associator and twist, with no extra requirement on F . The previousproposition shows that w-bialgebras arise naturally when one tries to solve them for agiven associator Φ of a weak quasi-bialgebra A . This gives a cohomological motivation forregarding the associator of a w-Hopf algebra as trivial.The following corollary extends to w-Hopf algebras a property known for Hopf algebras,see, e.g., [122]. Corollary 6.15.
Let A be a w-Hopf algebra and F ∈ A ⊗ A a -cocycle. Then the element u F = m ◦ S ⊗ F − ) is invertible and u − F = m ◦ ⊗ S ( F ) .Proof. The twisted weak quasi-bialgebra A F is a w-bialgebra thanks to Prop. 6.13. If S isthe strong antipode of A then A F has weak quasi-Hopf algebra antipode ( S, α F , β F ) where α F = m ◦ S ⊗ F − ), β F = m ◦ ⊗ S ( F ), by (4.11). Hence we can apply Prop. 6.5 to A F and deduce that α F and β F are inverses of one another. (cid:3) Proposition 6.16. If F is a -cocycle of A and G is a -cocycle of A F then GF is a -cocycle of A . We introduce two examples of 2-cocycles that will be useful.
Proposition 6.17.
Let v ∈ A be an invertible element with ε ( v ) = 1 and F ∈ A ⊗ A a -cocycle, then F v := v ⊗ vF ∆( v − ) is a -cocycle as well.Proof. Obviously F − v = ∆( v ) F − v − ⊗ v − . A computation shows that the left hand sideof (6.7) equals1 ⊗ ∆ ◦ ∆( v )1 ⊗ ∆( F − ) I ⊗ F − F ⊗ I ∆ ⊗ F )∆ ⊗ ◦ ∆( v − ) =1 ⊗ ∆ ◦ ∆( v ) Q P ∆ ⊗ ◦ ∆( v − ) =1 ⊗ ∆ ◦ ∆( v )1 ⊗ ∆ ◦ ∆( v − ) Q P = Q P . Relation (6.8) for F v can be proved in a similar way. (cid:3) Proposition 6.18.
Let E ∈ A ⊗ A be an idempotent satisfying ε ⊗ E ) = 1 ⊗ ε ( E ) = I,EP E = E, P EP = P . Then F = EP defines a trivial twist with D ( F ) = P , R ( F ) = E and F − = P E. It is a -cocycle if and only if the following additional relations hold, Q ⊗ ∆( E ) I ⊗ EE ⊗ ⊗ E ) P = Q P ,P ∆ ⊗ I ( E ) E ⊗ II ⊗ E ⊗ ∆( E ) Q = P Q . We omit the proof as it follows from a simple computation.7.
Quasitriangular and ribbon structures
The notion of quasitriangular Hopf algebra was introduced by Drinfeld in [32] and ex-tended to the quasi-Hopf algebra case in [34]. In this section we introduce and studyquasitriangular structure for weak quasi-Hopf algebras. We shall then restrict to weakquasi-Hopf algebras with a strong antipode and introduce the notion of ribbon structurein this case. In particular, we develop the basic results for this special subclass. For someresults for which computational difficulties would arise, we further restrict to the specialsubclass of w-Hopf algebras. In this case, we are able to present arguments extending thecorresponding results for Hopf algebras. We conclude the section explaining how later onwe shall extend all the results of this section concerning w-Hopf algebras to weak quasi-Hopf algebras with a strong antipode. This extension will be useful for the forthcomingdevelopments of the paper of Sect. 21 and for our applications of Sect. 24, 18.With any weak quasi-bialgebra A , we associate the opposite algebra A op with data givenby ε op = ε, ∆ op ( a ) := σ ◦ ∆( a ) , Φ op := Φ − , (7.1)where σ is the transposition automorphism of A ⊗ A and Φ − understood in a partialsense. Note that A op is a w-bialgebra if so is A . Definition 7.1.
A quasitriangular structure on A , also referred to R -matrix axioms, isdefined by a partially invertible element R ∈ A ⊗ A , ( R ∈ M ( A ⊗ A ) if A is discrete)satisfying the following properties, D ( R ) = ∆( I ) , R ( R ) = ∆ op ( I ) (7.2)∆ op ( a ) = R ∆( a ) R − , (7.3)∆ ⊗ R ) = Φ R Φ − R Φ , (7.4)1 ⊗ ∆( R ) = Φ − R Φ R Φ − , (7.5)We follow the standard notation: for a simple tensor a = a ⊗ · · · ⊗ a n ∈ A ⊗ n anda permutation i ∈ P n , a i ...i n is the simple tensor having a j in the i j -th component. If a ∈ A ⊗ k with k < n then we apply this definition to a tensored on the right with n − k copies of the identity operator. Furthermore relations (7.2)–(7.5) imply the analogue ofthe Yang-Baxter relation, which, taking into account (7.4) and (7.5), can be written in thefollowing form Φ − = I ⊗ R ⊗ ∆( R )Φ∆ ⊗ R − ) R − ⊗ I. (7.6) EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 39
Relations (7.2), (7.3), (7.6), and the following property (7.15) express the twist relation A op = A R . (7.7)Given a ∗ -algebra A endowed with the structure of a weak quasi-bialgebra, we can formanother weak quasi-bialgebra ˜ A , the adjoint algebra with the same algebra structure butcounit, coproduct, and associator given by˜ ε ( a ) := ε ( a ∗ ) , ˜∆( a ) := ∆( a ∗ ) ∗ , ˜Φ := Φ ∗− . (7.8)Note that if B is a ∗ -algebra, and p and q are idempotents of B and if T ∈ ( p, q ) then T ∗ ∈ ( q ∗ , p ∗ ). Hence if T is partially invertible in ( p, q ), so is T ∗ in ( q ∗ , p ∗ ). We understandΦ ∗− in this way. It will be useful to observe that Proposition 7.2. If R is an R -matrix for A then a) R op := R is an R -matrix for A op , b) if A is a ∗ -algebra, ˜ R := R ∗− is an R -matrix for ˜ A , c) if F ∈ A ⊗ A is a twist, R F := F RF − is an R -matrix for A F , d) R − is another R -matrix for A . Definition 7.3.
By a quasitriangular w-bialgebra we understand a w-bialgebra endowedwith a quasitriangular structure as a weak quasi-bialgebra.Note that any R -matrix of a w-Hopf algebra is a 2-cocycle by (7.7). An importantproperty for representation theory of quasitriangular Hopf algebras is that the square of theantipode is an inner automorphism. This was shown by Drinfeld who explicitly constructedan implementing invertible element u ∈ A for Hopf algebras [33]. Furthermore, Reshetikhinand Turaev introduced the notion of ribbon Hopf algebra [108]. We next show that thesedevelopments have extensions to w-Hopf algebras, although the computations in the proofsbecome more involved. We start with the following remark giving a simplification of theaxioms in the w-Hopf algebra case. Proposition 7.4.
Equations (7.4) and (7.5) for a w-Hopf algebra are equivalent to ∆ ⊗ R ) = Φ R R Φ , (7.9)1 ⊗ ∆( R ) = Φ − R R Φ − . (7.10) Proof.
We prove (7.9). We have Φ = 1 ⊗ ∆( P )∆ ⊗ P ), Φ − = ∆ ⊗ P )1 ⊗ ∆( P ),Φ = ∆ ⊗ P ′ ) a ⊗ b ⊗ a , and Φ − = a ⊗ b ⊗ a ⊗ ∆ op ( P ) where P = ∆( I ), P ′ = ∆ op ( I ),and we have used the notation ∆( b ) = b ⊗ b and P = a ⊗ b . By (7.3) we have R Φ − R = R a ⊗ b ⊗ a ⊗ ∆ op ( P ) R = a ⊗ b ⊗ a R R ⊗ ∆( P )and the conclusion follows. For (7.10) we similarly have Φ − = 1 ⊗ ∆( P ′ ) b ⊗ a ⊗ b andΦ = b ⊗ a ⊗ b ∆ op ⊗ P ). (cid:3) We give a definition of ribbon weak quasi-Hopf algebra A with a strong antipode ex-tending the corresponding notion for Hopf algebras due to [107]. Definition 7.5.
Let A be a (discrete) weak quasi-bialgebra Then A is called balanced ifit is quasitriangular and is endowed with an invertible central element v ∈ A ( v ∈ M ( A ))such that R R = v ⊗ v ∆( v − ) , (7.11)where R is the R -matrix. If in addition A has an antipode ( S, α, β ) such that S ( v ) = v ,then A is called a ribbon weak quasi-Hopf algebra , and v the ribbon element . A balanced(ribbon) w-bialgebra is a w-bialgebra (w-Hopf algebra) is defined in the natural way.Note that the definition does not depend on the choice of the antipode by Prop. 4.9.We next introduce Drinfeld element u . For simplicity, we restrict to the case of a weakquasi-Hopf algebra with strong antipode. This will suffice for our applications. Definition 7.6.
Let A be a quasitriangular weak quasi-Hopf algebra with strong antipode S and R -matrix R . The element u = X i S ( t i ) r i (7.12)where R = P i r i ⊗ t i is called Drinfeld element . We also set R − = P j r j ⊗ t j . Proposition 7.7.
Let A be a quasitriangular weak quasi-Hopf algebra with strong antipode S and u the associated Drinfeld element. Then u is invertible, u − = P j S − ( t j ) r j and S ( x ) = uxu − , x ∈ A. (7.13) Proof.
This proof is a generalisation of the corresponding proof for quasitriangular Hopfalgebras, see e.g. [73]. In the following computations we use the notation ∆( x ) = x ⊗ x for x ∈ A , ∆( I ) = a ⊗ b , R = r ⊗ t , Φ − = x ′ ⊗ y ′ ⊗ z ′ . We have∆ op ⊗ x )) R ⊗ I Φ − = R ⊗ I Φ − ⊗ ∆(∆( x ))that accordingly may be written as x , rx ′ ⊗ x , ty ′ ⊗ x z ′ = rx ′ x ⊗ ty ′ x , ⊗ z ′ x , . Applying 1 ⊗ S ⊗ S and multiplying from right to left gives by (4.7), (4.2), S ( x ) w = wx, w := S ( z ′ ) S ( y ′ ) ux ′ . (7.14)The 3-cocycle relation Φ − ⊗ I = ∆ ⊗ ⊗ − )1 ⊗ ⊗ ∆(Φ − ) I ⊗ Φ1 ⊗ ∆ ⊗ w = u . The last argument extends in a straightforward way the case of quasi-Hopfalgebras, see the proof of Lemma 2.4 in [17]. The formula for u − follows from Cor. 4.10b). (cid:3) Note that this proposition does not depend on the R -matrix properties (7.4), (7.5).But when they do hold, we obtain stronger relations for u in a way that extends thecorresponding relations for quasitriangular Hopf algebras. The following extends Lemma2.1.1, Ch. XI, of [122], or Theorem VIII.2.4 of [73] to w-Hopf algebras. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 41
Proposition 7.8. If A is a quasitriangular weak quasi-bialgebra algebra defined by R then ε ⊗ R ) = I, ⊗ ε ( R ) = I. (7.15) If A is a w-Hopf algebra, S ⊗ S ( R ) = f Rf − , (7.16) where f is the element defined in Prop. 4.12.Proof. The proof of (7.15) goes as in the bialgebra case, it suffices to apply ε ⊗ ⊗ ⊗ ⊗ ε to (7.4) and (7.5) respectively. To show (7.16) we tensor both sides of (7.9) bythe identity operator I on the left and multiply by I ⊗ ⊗ ∆( P )∆ ⊗ ∆( P ) on the rightand obtain 1 ⊗ ∆ ⊗ I ⊗ R )1 ⊗ ⊗ ∆( I ⊗ P )∆ ⊗ ∆( P ) = XR (7.17)where X = I ⊗ Φ R ⊗ ⊗ σ [1 ⊗ ⊗ ∆( I ⊗ P )∆ ⊗ ∆( P )] ,σ : A ⊗ A → A ⊗ A is the flip automorphism and we have used the intertwining relations(7.2), (7.3). We next recall from the first section the map V ( a ⊗ b ⊗ c ⊗ d ) = S ( b ) c ⊗ S ( a ) d that we wish to apply to both sides of (7.17) and we obtain f = V ( X ) R. (7.18)To show the claim we perform computations taking into account the following facts: a)one of the two ways the element f is defined for a weak quasi-Hopf algebra with strongantipode is f = V ( I ⊗ Φ − ⊗ ⊗ ∆(Φ)). For a w-Hopf algebra we have I ⊗ Φ − ⊗ ⊗ ∆(Φ) = I ⊗ ∆ ⊗ P )1 ⊗ ⊗ ∆(1 ⊗ ∆( P ))∆ ⊗ ∆( P ) =1 ⊗ ∆ ⊗ ⊗ ∆( P )) I ⊗ ⊗ ∆( P )∆ ⊗ ∆( P ) . b) We have V (1 ⊗ ∆ ⊗ Z ) Y ) = V ( Y ) as soon as m ◦ S ⊗ ε ⊗ Z ) = I , where m : A ⊗ A → A is the multiplication map. This holds in particular for Z = I ⊗ R and Z = 1 ⊗ ∆( P ),by (7.15) and (4.7). Hence the image of the left hand side of (7.17) under V is f . c) V ( XR ) = V ( X ) R . We next apply a similar procedure to relation (7.17) for the oppositew-Hopf algebra getting the relation1 ⊗ ∆ op ⊗ I ⊗ R )1 ⊗ ⊗ ∆ op ( I ⊗ P )∆ op ⊗ ∆ op ( P ) = X op R (7.19)where X op = I ⊗ Φ − R ⊗ ⊗ ∆( I ⊗ P )∆ op ⊗ ∆( P ) but now we apply the map W := σ ◦ S ⊗ S ◦ V op to both sides of (7.19), where V op acts as V but with S − inplace of S . To perform these computations we remark that: d) for the left hand sidewe use the identity S ⊗ S ◦ V op = V ◦ τ , where τ is the automorphism of A ⊗ taking a ⊗ a ⊗ a ⊗ a → a ⊗ a ⊗ a ⊗ a . e) the image of the left hand side of (7.19) under τ is1 ⊗ ∆ ⊗ R ⊗ I )∆ ⊗ ⊗ P ⊗ I )∆ ⊗ ∆( P )f) the second way in which f can be computed is f = V (Φ ⊗ I ∆ ⊗ ⊗ − )), and recallthat this was due to the 3-cocycle relation of Φ and the previous remark b). For a w-Hopfalgebra, computations similar to those in a) giveΦ ⊗ I ∆ ⊗ ⊗ − ) = 1 ⊗ ∆ ⊗ ⊗ P ))∆ ⊗ ⊗ P ⊗ I )∆ ⊗ ∆( P ) . Hence using b) again, the image of the left hand side of (7.19) under W is f . For theright hand side, we write W in the form W = V ◦ σ ⊗ σ ◦ τ . Simple computations showthat if α = σ ⊗ σ ◦ τ then α ( R ) = R and that V ( Y R ) = S ⊗ S ( R ) V ( Y ) for Y ∈ A ⊗ .Summarizing, the image of (7.19) under W is f = S ⊗ S ( R ) V ( α ( X op )) . (7.20)Comparing (7.18) and (7.20), the proof of (7.16) will be complete provided V ( X ) = V ( α ( X op )). To show this, a computation relying on by (7.2), (7.3), (4.4) gives I ⊗ Φ R ⊗ ⊗ σ [1 ⊗ ∆ ⊗ ⊗ ∆( P ))] = 1 ⊗ ∆ ⊗ Z ) I ⊗ Φ R where Z = 1 ⊗ ∆ op ( P ). It follows, by a), and the 3-cocycle relation, and (7.2), (7.3) again,1 ⊗ ∆ ⊗ Z ) X = I ⊗ Φ R ⊗ ⊗ σ [ I ⊗ Φ − ⊗ ⊗ ∆(Φ)] = I ⊗ Φ R ⊗ ⊗ σ [1 ⊗ ∆ ⊗ ⊗ I ∆ ⊗ ⊗ − )] =( I ⊗ Φ1 ⊗ ∆ ⊗ R ⊗ ⊗ σ [Φ ⊗ I ∆ ⊗ ⊗ − )] . On the other hand, α ( X op ) = Φ − R ⊗ ⊗ σ [∆ ⊗ P ) ⊗ I ∆ ⊗ ∆( P )]and similar computations give a , ⊗ a ⊗ b ⊗ a , α ( X op ) = Φ − R ⊗ ⊗ σ [Φ ⊗ I ∆ ⊗ ⊗ − )] . Hence 1 ⊗ ∆ ⊗ Z ) X = ( I ⊗ Φ1 ⊗ ∆ ⊗ ⊗ I ) a , ⊗ a ⊗ b ⊗ a , α ( X op ) =(1 ⊗ ⊗ ∆(Φ)∆ ⊗ ⊗ α ( X op )by the 3-cocycle relation again. It now suffices to apply V on both sides of this identity. (cid:3) Proposition 7.9.
Drinfeld element u of a quasitriangular w-Hopf algebra satisfies R R ∆( u ) = ∆( u ) R R = f − S ⊗ S ( f ) u ⊗ u. Proof.
The first equality follows easily from (7.3). We show the second equality. The lefthand side equals, by Prop. 4.12,∆( u ) R R = ∆( S ( t )) R R ∆( r ) = f − S ⊗ S (∆ op ( t )) f R R ∆( r ) . where notation is as before: R = r ⊗ t , ∆( I ) = P = a ⊗ b , ∆( x ) = x ⊗ x . We are thusreduced to show the equality S ⊗ S (∆ op ( t )) f R R ∆( r ) = S ⊗ S ( f ) u ⊗ u. (7.21)We denote by λ and ρ the left and right hand sides of (7.21), respectively. We use again thethe map V : A ⊗ → A ⊗ , V ( a ⊗ b ⊗ c ⊗ d ) = S ( b ) c ⊗ S ( a ) d , and recall that f = V ( A ) = V ( X ),where we have set A = I ⊗ Φ − ⊗ ⊗ ∆(Φ) and X = Φ ⊗ I ∆ ⊗ ⊗ − ). We shall alsoneed the property V ( La ⊗ b ⊗ c ⊗ d ) = S ( b ) ⊗ S ( a ) V ( L ) c ⊗ d. (7.22)For example, it shows that λ = V ( A · ∆( t ) ⊗ [ R R ∆( r )]) . EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 43
Furthermore, assuming that Y ∈ A ⊗ satisfies V ( Y ) = u ⊗ u , and writing X = x ⊗ y ⊗ w ⊗ z ,we have ρ = S ⊗ S (( S ( y ) w ⊗ S ( x ) z ) ) u ⊗ u = S ( z ) S ( x ) u ⊗ S ( w ) S ( y ) u = S ( z ) ux ⊗ S ( w ) uy = V ( Y w ⊗ z ⊗ x ⊗ y ) = V ( Y X ) . We start computing λ . By (7.9), and (7 . R R ∆( r ) ⊗ ∆( t ) = R R ∆ ⊗ ⊗ − R R Φ − ) =∆ ⊗ ⊗ − ) R ∆ op ⊗ ⊗ R ) R ∆ ⊗ ⊗ R )∆ ⊗ ⊗ − ) =∆ ⊗ ⊗ − ) R Φ R R Φ R Φ R R Φ ∆ ⊗ ⊗ − ) . After applying the permutation of (13)(24) ∈ P , and taking into account A ⊗ ⊗ ∆(Φ − ) = I ⊗ Φ − ⊗ ⊗ ∆(1 ⊗ ∆( P )) = 1 ⊗ ∆ ⊗ ⊗ ∆( P )) I ⊗ Φ − we see that λ equals V (1 ⊗ ∆ ⊗ ⊗ ∆( P ))[ I ⊗ Φ − R Φ R R Φ ][ R Φ R R Φ ]1 ⊗ ⊗ ∆(Φ − )) . The first bracketed element is the shift to the right of Φ − R Φ R R Φ , and computa-tions similar to those of Prop. 7.4 show that the latter equals (1 ⊗ ∆( R )) R b ⊗ a ⊗ a .Similarly, the second bracketed element acts as identity on the second factor, and as a ⊗ b ⊗ b R R Φ − R Φ in the remaining factors. This in turn equals a ⊗ b ⊗ b (1 ⊗ ∆( R )) R Φ by Prop. 7.4 again and property (6.5). Hence λ equals V (1 ⊗ ∆ ⊗ ⊗ ∆( P ))(1 ⊗ ∆( R )) R I ⊗ b ⊗ a ⊗ a · a ⊗ I ⊗ b ⊗ b (1 ⊗ ∆( R )) R X ) . Taking the range of X into account, we are finally left to show that Y := 1 ⊗ ∆ ⊗ ⊗ ∆( P ))(1 ⊗ ∆( R )) R I ⊗ b ⊗ a ⊗ a · a ⊗ I ⊗ b ⊗ b (1 ⊗ ∆( R )) R (1 ⊗ ∆ ⊗ ⊗ P )) . indeed satisfies V ( Y ) = u ⊗ u . To this aim, we move the two idempotents at both extremestowards the center using the commutation relations (7.3) and (6.4) and the domain relations(6 . Y = (1 ⊗ ∆( R )) R I ⊗ b ⊗ b ⊗ a (1 ⊗ ∆ ⊗ ⊗ ∆( P )) · (1 ⊗ ∆ ⊗ ⊗ P )) a ⊗ I ⊗ b ⊗ a (1 ⊗ ∆( R )) R =(1 ⊗ ∆( R )) R (1 ⊗ ∆ ⊗ (1 ⊗ ∆( R )) R . Now 1 ⊗ ∆ ⊗ ⊗ ∆( P ))(1 ⊗ ∆( R )) R and (1 ⊗ ∆( R )) R (1 ⊗ ∆ ⊗ ⊗ P )) have( Q ) and ( P ) as domain and range respectively, and furthermore Q ⊗ ∆ ⊗ P = Q ∆ ⊗ ∆( P ) P by the cocycle relation, see Remark 6.3. Hence we can also write Y = (1 ⊗ ∆( R )) R (∆ ⊗ ∆( P )) (1 ⊗ ∆( R )) R . We are now able to compute V ( Y ) by means of an iterative use of (7.22): V ((1 ⊗ ∆( R )) ) = I ⊗ I,V ((1 ⊗ ∆( R )) R ) = u ⊗ I,V ((1 ⊗ ∆( R )) R (∆ ⊗ ∆( P )) ) = S ( b ) ⊗ S ( a ) · u ⊗ I · b ⊗ a = S ( b ) ub ⊗ S ( a ) a = S ( S ( b ) b ) u ⊗ ε ( a ) = ε ( b ) ⊗ ε ( a ) u ⊗ I = u ⊗ I,V ((1 ⊗ ∆( R )) R (∆ ⊗ ∆( P )) (1 ⊗ ∆( R )) ) = I ⊗ S ( t ) · u ⊗ I · r ⊗ t = u ⊗ I ⊗ ε ( R ) = u ⊗ I,V ( Y ) = u ⊗ u, and the proof is complete. (cid:3) At the level of representation theory, the previous proposition establishes commutativityof the following diagram. ρ ⊗ σ ρ ∨∨ ⊗ σ ∨∨ σ ⊗ ρ ρ ⊗ σ ( ρ ⊗ σ ) ∨∨ u ρ ⊗ u σ ε ( ρ,σ ) ε ( σ,ρ ) u ρ ⊗ σ Remark 7.10.
It follows from (7.11) and (7.15) that if v makes A balanced then ε ( v ) = 1.Furthermore when v is a ribbon element, applying m ◦ S ⊗ v = uS ( u ), with u Drinfeld element, as in Def. 7.6.
Corollary 7.11.
The elements u and v of a ribbon w-Hopf algebra satisfy ∆( uv − ) = f − S ⊗ S ( f ) uv − ⊗ uv − . Remark 7.12.
Altschuler and Coste extended ribbon structures to quasi-Hopf algebras [1],stated analogues of the lemmas of this section and outlined some of the proofs. Completeproofs have been given in [60, 17]. In this passage, the construction of Drinfeld element u and the notion of ribbon quasi-Hopf algebra needs to be suitaby modified. Moreover, theproof of the analogue of Prop. 7.8, Prop. 7.9 become more involved. Likely, these workstogether with the results of this section lead to extensions of the main properties of ribbonstructures to the more general setting of weak quasi-Hopf algebras. However, we shallrefrain from doing this, and rather take an alternative categorical approach. More in detail,motivated also by the study of quantum dimension, in Sect. 20 we shall revisit Drinfeldisomorphism and ribbon structures in the framework of tensor categories. Moreover, weshall study more general structures (coboundary symmetries). In particular, it will followfrom the results of that section that via Tannaka-Krein duality when A is a discrete weakquasi-Hopf algebra with a strong antipode then Drinfeld element is still defined as in Def.7.6. It will also follow that all the special results of this section concerning w-Hopf algebrasextend to this setting with the same statements, and this will suffice for the forthcomingdevelopments of our paper considered in Sect. 21, and for our applications of Sects. 24,18. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 45
8. Ω -involution and C*-structure
In this section we introduce a ∗ -involution ∗ : A → A to a weak quasi-Hopf algebra. Inthe usual approach, among the compatibility conditions with the weak quasi-Hopf algebrastucture, one requires for example that the involution and the coproduct commute. Weshall relax these compatibility conditions via the introduction of a twist Ω which is part ofthe axioms of the involutive structure. There are several reasons to study such structures.On one hand, unlike the ordinary approach, the more general notion is invariant underDrinfeld twist operation A → A F . Another motivation for us arises from consideringnatural examples, which include the Drinfeld-Jimbo quantum groups U q ( g ) for the valuesof the deformation parameter q with | q | = 1. Finally, as we shall see more precisely inSect. 10, Ω-involutions of weak quasi-Hopf algebras describe unitary structures in fusioncategories and intervene in the study of tensor ∗ -equivalences. Definition 8.1.
A weak quasi bialgebra A will be called Ω -involutive if it is endowed witha ∗ -involution ∗ : A → A making it into a ∗ -algebra and a selfadjoint twist Ω ∈ A ⊗ A suchthat ˜ A = A Ω , with ˜ A the adjoint weak quasi bialgebra defined in (7.8). Explicitly, thismeans that Ω ∈ A ⊗ A is a partially invertible element satisfyingΩ ∗ = Ω , (8.1) D (Ω) = ∆( I ) , R (Ω) = ∆( I ) ∗ , (8.2)∆( a ∗ ) = Ω − ∆( a ) ∗ Ω , a ∈ A, (8.3) ε ⊗ I = 1 ⊗ ε (Ω) (8.4)Φ ∗− = I ⊗ Ω1 ⊗ ∆(Ω)Φ∆ ⊗ − )Ω − ⊗ I (8.5)A unitary weak quasi bialgebra is an Ω-involutive weak quasi bialgebra such that A is aC*-algebra and Ω is positive in A ⊗ A . Corresponding Hopf versions assume the existenceof an antipode S . Note that in general we require no compatibility assumption with theinvolution.The most important relations are the intertwining property with the coproduct (8.3)and the compatibility relation (8.5) between ( ∗ , Ω) and the associator.The notion of Ω-involution for a semisimple weak quasi Hopf algebra is the most generalinvolutive structure that gives rise to a tensor ∗ -category structure on the category of finitedimensional representations of A . For example, we shall see that every fusion tensor ∗ -category ( C ∗ -category) arises from a semisimple Ω-involutive (unitary) weak quasi Hopfalgebra. We next recall several well known and important special notions. Remark 8.2. a) A is a Hopf ∗ -bialgebra precisely when ∆( I ) = I ⊗ I and Ω = I ⊗ I ,Φ = I ⊗ I ⊗ I . These structures are widely studied when A is a C ∗ -algebra in the operatoralgebraic approach to quantum groups see e.g. [100], [119]. b) When A is a bialgebra(∆( I ) = I ⊗ I , Φ = I ⊗ I ⊗ I ) (8.1) and (8.2) say that Ω is a selfadjoint invertibleelement. Note that in this case (8.5) says that Ω is a 2-cocycle in the usual sense forHopf algebras. In the next proposition we discuss an extension of this property to weakbialgebras. c) If A is as in b) and A is a C ∗ -algebra with Ω positive then the twisted algebra A F , with F = Ω / is a quasi C*-bialgebra in the sense of a). We shall shortlyconsider an extension of the notion of triviality of Ω in the weak quasi bialgebras which isthe algebraic counterpart of the notion of unitary weak quasi-tensor functor of Def. 2.14. d) When A is a quasi-bialgebra (that is ∆( I ) = I ⊗ I and Φ non-trivial) we recover thenotion introduced by Gould and Lekatsas [52]. Example 8.3.
The Hopf algebras U q ( g ) for | q | = 1 considered by Wenzl in [128] are for usimportant examples of Ω-involutive Hopf algebras with a non-trivial selfadjoint 2-cocycle Ωin the sense of part b) of the previous remark. We shall discuss these examples in Section24. In this case, Ω is canonically induced by the R -matrix . Furthermore, in Sections21, 23, 24 we shall construct new examples of semisimple Ω-involutive or unitary w-Hopfalgebras associated to U q ( g ) for q a suitable root of unity, corresponding to the associatedunitary fusion categories.We next extend the 2-cocycle property of Ω from bialgebras to w-bialgebras. Proposition 8.4. If ( ∗ , Ω) makes a w-bialgebra ( A, ∆ , ε, Φ = Q P ) Ω -involutive then Ω is a -cocycle.Proof. By definition Φ = Q P is an associator with Φ − = P Q , see Sect. 6. Then( A, ˜∆ , ε, ˜Φ) is a w-Hopf algebra as well since˜Φ = ( Q P ) ∗− = ( Q P ) − ∗ = ( P Q ) ∗ = Q ∗ P ∗ = 1 ⊗ ˜∆( ˜∆( I )) ˜∆ ⊗
1( ˜∆( I ))and similarly ˜Φ − = ˜∆ ⊗
1( ˜∆( I ))1 ⊗ ˜∆( ˜∆( I )). By (8.5) and Prop. 6.13 we see that Ω is a2-cocycle. (cid:3) Definition 8.5.
Let A be a discrete algebra in the sense of Def. 4.16. A positive ∗ -involution on A is a ∗ -involution such that A can be completed to a C*-algebra. Wemay then identify A with an algebraic direct sum of matrix subalgebras with the usual ∗ -involution. An Ω-involutive structure on A is defined as in the unital case but Ω ishere allowed to be a (selfadjoint) element in M ( A ⊗ A ). A unitary discrete weak quasi-Hopf algebra is defined by further requiring that Ω has positive components in the matrixsubalgebras. In the particular case where A is a w-Hopf algebra, we shall refer to A as a unitary discrete w-Hopf algebra .Unless otherwise stated, involutions of discrete algebras will be assumed positive. Thiswill hold for most part of this paper. We next describe the Ω-involutions on a simple classof discrete algebras. To be precise, the Ω-involution of U q ( g ) is not comprised in Def. 8.1. This is due to the fact thatthe R -matrix lies in a suitable topological completion of U q ( g ) ⊗ U q ( g ). However, when we consider thecategory of finite dimensional representations of U q ( g ), this inconvenience is not source of complicationsin that it gives rise to a braided tensor category, as explained in [114]. It follows that the associated Ω alsolies in the completed algebra. Similarly to the R -matrix case, in this paper we will consider applicationsof the notion of Ω-involution to categories of finite dimensional representations, see Sect. 9–17, and weshall refrain from giving a more general definition of Ω-involution. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 47
Example 8.6.
We consider the bialgebra C ω ( G ) of complex valued functions of a finitegroup G with the usual coproduct and associator given by a T -valued 3-cocycle ω , seeEx. 5.12. A natural unitary structure is given by the C*-structure of C ω ( G ) and Ω = I .More generally, a general Ω-involution for C ω ( G ) over the same C*-algebra is given bya normalized 2-cocycle Ω( g, h ) with values in R × , that is a function satisfying Ω(1 , g ) =Ω( g,
1) = 1 and Ω( g, h )Ω( gh, k ) = Ω( h, k )Ω( g, hk ) for all g , h , k ∈ G . The correspondingquasi-Hopf algebra is unitary if and only of Ω( g, h ) > g , h ∈ G . This is not alwaysthe case, an example is given by G = Z Ω( g, g ) = − g the group generator.In the next sections we shall see examples of unitary discrete weak quasi Hopf algebrasarising from unitary tensor categories, Sects. 10, and quantum groups Sects. 21, 23, 24.Moreover we shall discuss conditions which guarantee unitarity, see Theorem 19.2. In thefollowing proposition we show that the fact that ˜ A and A Ω have the same counit is aredundant assumption. Proposition 8.7.
The counit ε of a weak quasi-bialgebra A is unique. If A is a weakquasi-Hopf algebra with antipode S the counit satisfies ε ◦ S = ε . If A is an Ω -involutiveweak quasi-bialgebra then ε ( a ∗ ) = ε ( a ) , for every a ∈ A .Proof. The first two statements can be proved in the same way as for quasi-bialgebras,namely the first follows from (4.2) while the second from applying the counit to one of theequations (4.7). For the last statement it suffices to show that e ε ( a ) := ε ( a ∗ ) is a counit.For example, (1 ⊗ e ε )(∆( a )) = a (1) e ε ( a (2) ) =( a ∗ (1) ε ( a ∗ (2) )) ∗ = (1 ⊗ ε (∆( a ) ∗ )) ∗ =(1 ⊗ ε (Ω∆( a ∗ )Ω − )) ∗ = (1 ⊗ ε (∆( a ∗ ))) ∗ = a. (cid:3) Proposition 8.8. a) Let A be an Ω -involutive weak quasi-bialgebra and F ∈ A ⊗ A atwist (Def. 4.7). Then A F is an Ω F -involutive weak quasi-bialgebra with the sameinvolution as A and Ω F := F − ∗ Ω F − , (Ω F ) − := F Ω − F ∗ (8.6)b) If A is a discrete pre-C*-algebra and Ω is positive in M ( A ⊗ A ) then Ω F is positiveas well. We discuss a useful application of the twist of the unitary structure.
Definition 8.9.
Let A be a weak quasi bialgebra with a ∗ -involution. An Ω-involutioncompatible with ∗ on A is called trivial if it is given by Ω = ∆( I ) ∗ ∆( I ) and Ω − =∆( I )∆( I ) ∗ . Thus Ω is a trivial twist. We shall call it strongly trivial if in addition ∆( I ) isselfadjoint, that is equivalent to require that commutes ∆ commutes with the ∗ -involutionas in the usual ∗ -bialgebra theory. In this case, ∆( I ) is a selfadjoint projection. With a strongly trivial involution, ∆ commutes with ∗ and the associator Φ satisfiesΦ ∗ = Φ − . The above notions of (strong) triviality has the same motivation as that of andare related to those of (strongly) unitary weak quasi tensor functor discussed before Def.2.14. Remark 8.10. a) As in the case of weak quasitensor structures, when A is a weak quasibialgebra with a ∗ -involution and a trivial Ω-involution compatible with ∗ then T = ∆( I )is a twist with left inverse T − = ∆( I )∆( I ) ∗ (or T ′ = ∆( I ) ∗ ∆( I ) with T ′− = ∆( I )) givinga new wqh A T ( A T ′ ) with strongly trivial involution. b) When A is a discrete unitary weakquasi-bialgebra with a trivial Ω-involution then this involution is automatically stronglytrivial. This follows from the fact that we are in a C ∗ -setting, Prop. 2.16 and the followingTannaka-Krein duality, Theorem 10.5. Example 8.11.
We have the following generalization of the construction in Remark c)in 8.2. Let A be a unitary discrete weak quasi bialgebra with an Ω-involution given byΩ ∈ M ( A ⊗ A ). We may consider T = Ω / defined via continuous functional calculus ineach full matrix subalgebra of M ( A ⊗ A ). This element satisfies the properties T ∆( I ) = T ,∆( I ) ∗ T = T , and ω ⊗ T ) = 1 ⊗ ω ( T ) = 1, so we may regard T as an element of A with the same domain ∆( I ) as Ω. Applying the same construction to Ω − , we construct T ′ = (Ω − ) / ∈ M ( A ⊗ A ) with range ∆( I ). Corollary 8.12.
Let A be a discrete unitary weak quasi bialgebra defined by Ω and assumethat (Ω − ) / Ω / = ∆( I ) . Let us regard T = Ω / as a twist with left inverse T − =(Ω − ) / . Then the twisted Ω -involution of A T is trivial, and therefore strongly trivial.Proof. By part b) of Prop. 8.8, Ω T = ∆ T ( I ) ∗ ∆ T ( I ) and Ω − T = ∆ T ( I )∆ T ( I ) ∗ . Strongtriviality follows again from the fact that we are in a C ∗ -setting, Prop. 2.16 and Tannaka-Krein duality Theorem 10.5. (cid:3) We shall refer to A Ω / as the unitarization of A . We next introduce a deformation of anΩ-involution on a given weak bialgebra that may be thought of as analogous to the twistoperation for the weak quasi bialgebra structure. Definition 8.13.
Let A be an Ω-involutive weak quasi bialgebra A defined by ( ∗ , Ω). Atwist for the involutive structure is an invertible selfadjoint t ∈ A such that ε ( t ) = 1. If A is discrete in the sense of Def. 8.5 we allow t ∈ M ( A ). Proposition 8.14.
A twist t of an involution ( ∗ , Ω) gives rise to another involutive struc-ture on the same weak quasi bialgebra via a † := t − a ∗ t, Ω t := t − ⊗ t − Ω∆( t ) . If A is a C*-algebra under ∗ , or else if A is discrete, and ( ∗ , Ω) is a positive involution,then the same holds for A with respect to ( † , Ω t ) for any positive twist t .Proof. The proof of the first statement follows from routine computations. For example,Ω † t = Ω t follows from (8.3). We show the second statement. If k a k denotes a C*-norm of EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 49 A compatible with ∗ then k a k t := k t / at − / k is another C*-norm on A compatible with † . (Note that the original and the deformed norms are equivalent, hence completeness ofone is equivalent to completeness of the other.) Furthermore if Ω is positive with respectto the original involution, the element Ξ := t − / ⊗ t − / Ω / ∆( t / ) satisfies Ξ † Ξ = Ω t , soΩ t is positive with the † -involution of A ⊗ A . (cid:3) In the discrete case, any other involution making A into a pre-C*-algebra is of the kind a † = t − a ∗ t , with t determined up to a normalized central positive element of M ( A ). Thisimplies e the following useful result. Corollary 8.15.
If a discrete weak quasi bialgebra A can be made unitary with respect toan assigned pre-C*-algebra involution of A , the same is true for any other such involution. As for twists of bialgebra structures, twists of involutive structures admit a categoricalinterpretation, that will be discussed in Prop. 10.1. The next results exploit the relationsbetween antipode and Ω-involution.
Proposition 8.16.
Let ( S, α, β ) be an antipode of an Ω -involutive weak quasi-Hopf algebra.There is an invertible ω ∈ A such that S ( a ) = ωS − ( a ∗ ) ∗ ω − , a ∈ A, (8.7) S − ( β ) ∗ = ω − α Ω , S − ( α ) ∗ = β Ω ω (8.8) uniquely determined by (8.7) and one of (8.8). In particular when S is a strong antipodethen ω = m ( S ⊗ − )) , ω − = m (1 ⊗ S (Ω)) . (8.9) Proof.
The adjoint weak quasi-bialgebra ˜ A defined in (7.8) has antipode ( ˜ S, ˜ α, ˜ β ) with˜ S ( a ) := S − ( a ∗ ) ∗ , ˜ α := S − ( β ) ∗ , ˜ β := S − ( α ) ∗ . On the other hand, ˜ A = A Ω , and thereforeit also admits ( S Ω , α Ω , β Ω ) as an antipode by Prop. 4.8. The first statement follows fromProp. 4.9 and the last from a computation and (4.11). (cid:3) Corollary 8.17.
The following are equivalent for an antipode ( S, α, β ) , a) S commutes with ∗ , b) S − commutes with ∗ , c) S ( a ) = ωaω − , a ∈ A .If these conditions hold then ω ∗ ω and S ( ω ) ω are central. We next study dependence of the element ω introduced in Prop. 8.16 on twisting. Proposition 8.18.
Let A be an Ω -involutive weak quasi-Hopf algebra with antipode ( S, α, β ) and involutive structure ( ∗ , Ω) and associated element ω as in Prop. 8.16. a) Let (Ad( u ) S, uα, βu − ) be another antipode of A . The corresponding element isgiven by ω u = uωS − ( u ) ∗ . b) Let F ∈ A ⊗ A be a twist and consider the weak quasi Hopf algebra A F with antipode ( S, α F , β F ) and involutive structure ( ∗ , Ω F ) . Then the corresponding element isgiven by ω F = ω. Proof. a) follows from a computation. b) By the uniqueness stated in Prop. 8.16 we onlyneed to verify that S − ( β F ) ∗ = ω − ( α F ) Ω F . The claim follows in a straightforward wayfrom a computation based on (8.7) and the first relation in (8.8) which takes into accountthe definition of α F , β F in (4.11) and of Ω F in (8.6). (cid:3) Definition 8.19.
An Ω-involutive weak quasi-Hopf algebra is called of Kac type if itadmits a (unique) strong antipode satisfying one of the equivalent conditions stated inCor. 8.17.The definition is motivated by the fact that if A is in turn a Hopf ∗ –algebra in the usualsense (Ω = I ) then ω = I , and Cor. 8.17 reduces to the well known characterisation ofHopf ∗ –algebras of Kac type. Proposition 8.20.
Let A be a Hopf algebra such that ∆ op ( a ) ∗ = ∆( a ∗ ) , a ∈ A. (e.g. A is Ω -involutive and satisfies ∆ op ( a ) = Ω∆( a )Ω − for a ∈ A ). Then A is of Kactype.Proof. Since A is a Hopf algebra, it admits a unique strong antipode, denoted S . Further-more, our assumptions imply ∆( a ∗ ) = ∆ op ( a ) ∗ for a ∈ A . It follows that the antiautomor-phism ˜ S ( a ) := S ( a ∗ ) ∗ is another Hopf algebra antipode of A , as( m ◦ (1 ⊗ ˜ S ) ◦ ∆)( a ) = a (1) ˜ S ( a (2) ) = ( S ( a ∗ (2) ) a ∗ (1) ) ∗ = (8.10)[ m ◦ ( S ⊗ o p ( a ) ∗ )] ∗ = [ m ◦ ( S ⊗ a ∗ ))] ∗ = (8.11)( ε ( a ∗ ) I ) ∗ = ε ( a ) I. (8.12)Hence ˜ S = S by uniqueness. (cid:3) Wenzl shows in [128] that the assumptions of Prop. 8.20 are satisfied by the quantumgroups U q ( g ) for | q | = 1, cf. also Sect. 24. We shall extend Prop. 8.20 to w-Hopf algebrasendowed with a ∗ -involution and a strong antipode in Sect. 21, see Prop. 21.7.9. The categories
Rep h ( A ) and Rep + ( A )Let A be an Ω-involutive weak quasi-Hopf algebra. In this section we associate thecategory Rep h ( A ) of representations on non-degenerate Hermitian spaces, and we introducethe structure of a rigid tensor ∗ -category. Most importantly, the subclass of unitary weakquasi-Hopf algebras leads to rigid tensor C ∗ -categories Rep + ( A ).The basic notion is that of Hermitian space, that is a finite dimensional vector space V equipped with a non degenerate Hermitian form ( ξ, η ). If W is another such space, anylinear map T : V → W admits an adjoint T ∗ : W → V defined as in the more familiar caseof Hilbert spaces: ( T ξ, η ) = ( ξ, T ∗ η ). The category Herm with objects finite dimensionalHermitian spaces and morphisms linear maps between them is the simplest example of a ∗ -category. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 51
Definition 9.1.
Let A be a unital complex ∗ -algebra with involution ∗ : A → A . a) By a ∗ –representation we understand a representation ρ of A on a nondegenerate Hermitian space V ρ satisfying ρ ( a ∗ ) = ρ ( a ) ∗ for a ∈ A . b) A C*-representation of A is a ∗ -representation ona Hilbert space.The study of ∗ -representations on Hermitian spaces is motivated by U q ( g ), for | q | = 1[128]. In this case, Wenzl showed that for generic values of q , or for certain roots of unityof sufficiently high order there is a finite set of irreducible C*-representations [128]. In thelatter case representation theory is not semisimple. A brief review and connections withthe theory of representations of weak quasi-Hopf algebras will be studied in later sections.Let Rep h ( A ) be the category with objects ∗ –representations of A on nondegenerateHermitian spaces. If T ∈ ( ρ, σ ) is a morphism of Rep h ( A ), the adjoint map T ∗ : V σ → V ρ is still a morphism of Rep h ( A ). In this way Rep h ( A ) becomes a ∗ -category.An isometric morphism S ∈ ( ρ, σ ) between two ∗ -representations is a morphism satis-fying S ∗ S = 1. Similarly, a unitary is an invertible isometry, that is U ∗ U = 1, U U ∗ = 1.Therefore there is a natural notion of unitary equivalence between ∗ –representations ρ and σ . Unitary equivalence implies equivalence, but, unlike the case of Hilbert space ∗ –representations, the converse does not hold. In other words, a representation can be madeinto a ∗ -representation in more than one way, up to unitary equivalence. This can be seenwith the following simple construction.Given a ∗ -representation ρ , let ρ − denote the ∗ -representation with the same space andaction as ρ but with with the opposite Hermitian form: ( ξ, η ) V ρ − = − ( ξ, η ) V ρ . We shall referto ρ − as the opposite ∗ -representation . Note that ρ and ρ − are equivalent as representationsbut they are not unitarily equivalent in the following two cases, either ρ is irreducible, orit may reduce but it is a C*-representation. Indeed, given another ∗ -representation σ anda linear map T : V ρ → V σ with adjoint T ∗ with respect to the original forms, the adjoint of T as a map V ρ − → V σ or V ρ → V σ − is − T ∗ . Thus the unitarity condition for an intertwiner T : V ρ → V ρ − becomes T ∗ T = − I , with T ∗ the adjoint of T as a map V ρ → V ρ , and this isincompatible with either irreducibility ( T acts as a scalar) or the C*-assumption on ρ .A ∗ -representation σ is called an orthogonal direct sum of ρ and τ if there are isometries S ∈ ( ρ, σ ), T ∈ ( τ, σ ) such that SS ∗ + T T ∗ = 1. This implies that SV ρ and T V τ arespanning, orthogonal subspaces of V σ : ( SV ρ , T V τ ) = 0, and hence are complementaryby nondegeneracy of the form. We write σ = ρ ⊕ τ and refer to ρ and τ as orthogonalsummands of σ . If ρ and τ are ∗ –representations, the direct sum Hermitian form on V ρ ⊕ V τ makes this space into a ∗ -representation σ in the natural way and we have σ = ρ ⊕ τ viathe inclusions S : V ρ → V σ , T : V τ → V σ . Any other realisation of σ as a direct sum of ρ and σ will be unitarily equivalent to this.If A is not semisimple as an algebra, representations may admit invariant submoduleswhich are not summands. The following proposition shows that the ∗ –structure is usefulto distinguish between summands and submodules. Proposition 9.2. If S ∈ ( ρ, σ ) is an isometry in Rep h ( A ) , then E = SS ∗ is a selfadjointidempotent with range SV ρ , defining an orthogonal summand of σ . Conversely, everysubmodule W of V σ (i.e. a subspace of V σ invariant under all the σ ( a ) , a ∈ A ) for which the restricted Hermitian form is nondegenerate, is a ∗ –representation and an orthogonalsummand.Proof. In general, if the restriction of the Hermitian form of V σ is nondegenerate on asubmodule W then the adjoint of the restriction of an element σ ( a ) with respect to therestricted form equals the restriction of σ ( a ∗ ) by ∗ -invariance of σ and nondegeneracy.Hence W defines a ∗ -representation and the inclusion map S : W → V σ is an isometry.Given an isometry S ∈ ( ρ, σ ) in Rep h ( A ), E = SS ∗ obviously defines an algebraicsummand of σ . The ranges of E and 1 − E are orthogonal subspaces of V σ . This impliesthat the Hermitian form of V σ is nondegenerate on either subspace and therefore these are ∗ –representations ρ and τ such that σ = ρ ⊕ τ . (cid:3) We next give a criterion for nondegeneracy of Hermitian forms.
Proposition 9.3.
A nonzero Hermitian form on the vector space of an irreducible rep-resentation ρ of A making it ∗ -invariant is nondegenerate. Any other ∗ -representationstructure on ρ is unitarily equivalent to ρ or ρ − .Proof. The subspace V ⊥ ρ = { v ∈ V ρ , ( v, V ρ ) = 0 } is a submodule by ∗ -invariance of ρ , andit must be proper, hence V ⊥ ρ = 0 by irreducibility, and this shows nondegeneracy. Everyother nondegenerate Hermitian form on V ρ is defined by an invertible map B : V ρ → V ρ via ( ξ, η ) B = ( ξ, Bη ), with B selfadjoint with respect to the given Hermitian form. Theadjoint of a map T : V ρ → V ρ with respect to the new form as compared to the oldchanges to B − T ∗ B . The ∗ -invariance condition for ρ with respect to the new form readsas B − ρ ( a ∗ ) B = ρ ( a ∗ ) for a ∈ A by ∗ -invariance of ρ . Thus B is a nonzero real scalar. (cid:3) A tensor product of Hermitian spaces becomes an Hermitian space in the natural way:( ξ ⊗ ξ ′ , η ⊗ η ′ ) p := ( ξ, η )( ξ ′ , η ′ ). In this way Herm becomes a tensor ∗ -category, and it isthe unique ∗ -structure on Herm compatible with the tensor structure.We next describe how to obtain a tensor ∗ -category from an Ω-involutive weak quasi-bialgebra. Note that the ∗ -structure obtained restricting that of Herm to Rep h ( A ) isnot the correct one, as it does not make a tensor product of two ∗ –representations intoa ∗ -representation. This is due to the fact that the coproduct and ∗ -involution do notcommute. On the other hand, because of the twisted commutation relation they satisfy,one can consider a twist of the product form by the action of Ω,( ζ , ζ ′ ) Ω := ( ζ , Ω ζ ′ ) p , ζ , ζ ′ ∈ V ρ ⊗ ρ ′ , which is indeed a non degenerate and Hermitian form. Theorem 9.4.
Let A be an Ω -involutive weak quasi bialgebra. For every pair of ∗ –representations ρ , ρ ′ , the form ( · , · ) Ω on V ρ ⊗ V ′ ρ makes ρ ⊗ ρ ′ into a ∗ –representation. Inthis way Rep h ( A ) becomes a tensor ∗ –category with unitary associativity morphisms. Thiscategory is strict if A is a bialgebra. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 53
Proof.
Let V be a Hermitian space, and consider a new Hermitian form of V defined by agiven selfadjoint invertible A ∈ L ( V ). Denote by V A the associated Hermitian space. Let W , B , be another such pair. Given T ∈ L ( V, W ), we denote by T ∗ and T † the adjoint of T with respect to the new forms (that is as a map T : V A → W B ) and the original formrespectively. They are related by T ∗ = A − T † B. Therefore given T ∈ L ( V ρ ⊗ V ρ ′ , V σ ⊗ V σ ′ ),we have T ∗ = Ω − T † Ω with adjoints referred to the twisted form and the restricted productform respectively. Thus T ∗ = T † if T † commutes with the action of Ω. For example, thisalways holds for T = S ⊗ S ′ , with S ∈ ( ρ, σ ), S ′ ∈ ( ρ ′ , σ ′ ). Indeed, T † = S ∗ ⊗ S ′∗ , and S ∗ and S ′∗ are intertwiners. We at once find ( S ⊗ S ′ ) ∗ = S ∗ ⊗ S ′∗ . Notice that the productform is related to the involution of the tensor product ∗ –algebra A ⊗ A : ρ ⊗ ρ ′ ( b ∗ ) = ρ ⊗ ρ ′ ( b ) † , b ∈ A ⊗ A. Therefore for a ∈ A , ρ ⊗ ρ ′ ( a ) ∗ = ρ ⊗ ρ ′ (∆( a )) ∗ = ρ ⊗ ρ ′ (Ω − ) ρ ⊗ ρ ′ (∆( a )) † ρ ⊗ ρ ′ (Ω) = ρ ⊗ ρ ′ (Ω − ∆( a ) ∗ Ω) = ρ ⊗ ρ ′ (∆( a ∗ )) = ρ ⊗ ρ ′ ( a ∗ ) . Given ∗ -representations ρ , σ , τ , the ∗ –representations ( ρ ⊗ σ ) ⊗ τ and ρ ⊗ ( σ ⊗ τ ) act via themorphisms ∆ ⊗ ◦ ∆ and 1 ⊗ ∆ ◦ ∆, respectively, on the subspaces of V ρ ⊗ V σ ⊗ V τ determinedby the image of I under those morphisms. With respect to the triple product form, theassociated Hermitian forms are induced by Ω ⊗ I ∆ ⊗ I ⊗ Ω1 ⊗ ∆(Ω), respectively.To show that the associativity morphisms α ρ,σ,τ are unitary arrows of Rep h ( A ), we computetheir adjoints taking into account the remark at the beginning of the proof, α ∗ ρ,σ,τ = (Ω ⊗ I ∆ ⊗ − α † ρ,σ,τ I ⊗ Ω1 ⊗ ∆(Ω) = ρ ⊗ σ ⊗ τ (∆ ⊗ − )Ω − ⊗ I Φ ∗ I ⊗ Ω1 ⊗ ∆(Ω)) = ρ ⊗ σ ⊗ τ (Φ − ) = α − ρ,σ,τ . If in addition A is a bialgebra then Φ is the trivial associator, hence Ω is a 2-cocycle byProp. 8.4. This means that ( ρ ⊗ σ ) ⊗ τ and ρ ⊗ ( σ ⊗ τ ) also coincide as ∗ –representations.Since the associativity morphisms are trivial, Rep h ( A ) is strict. (cid:3) Corollary 9.5.
Suppose that A is a unitary weak quasi bialgebra A . Then the full subcat-egory Rep + ( A ) of Rep h ( A ) with objects C*-representations is a tensor C ∗ -category.Proof. The Ω-twisted inner product of a tensor product of two C*-representations is stilla positive inner product by positivity of Ω. (cid:3)
Proposition 9.6.
Let A be an Ω -involutive weak quasi bialgebra. The forgetful functor F : Rep h ( A ) → Herm (or F : Rep + ( A ) → Hilb in the C*-case) is a ∗ -functor. The naturaltransformations satisfy F ∗ ρ,σ = ρ ⊗ σ (Ω) ◦ G ρ,σ , G ∗ ρ,σ = F ρ,σ ◦ ρ ⊗ σ (Ω − ) . (9.1) Proof. ∗ -invariance of F is clear. Relations (9.1) follow from computations as in the proofof Theorem 9.4. (cid:3) We observe that thanks to G ρ,σ ◦ F ρ,σ = ρ ⊗ σ (∆( I )), relations (9.1) can also be writtenin the form F ∗ ρ,σ ◦ F ρ,σ = ρ ⊗ σ (Ω) , G ρ,σ ◦ G ∗ ρ,σ = ρ ⊗ σ (Ω − ) . (9.2) Proposition 9.7.
Let A be an Ω -involutive weak quasi-bialgebra with involution ( ∗ , Ω) and F ∈ A ⊗ A a twist. Consider the twisted algebra A F with involution ( ∗ , Ω F ) as in Prop. 8.8.Then the tensor equivalence E defined in Prop. 5.2 restricts to a unitary tensor equivalencebetween Rep h ( A ) → Rep h ( A F ) ( Rep + ( A ) → Rep + ( A F ) in the unitary case).Proof. The two algebras have the same ∗ -involution, hence the equivalence is a ∗ -functor.We show unitarity of the associated natural transformation, which is given by the actionof E ρ,σ = ρ ⊗ σ ( F − ) from E ( ρ ) ⊗ E ( σ ) to E ( ρ ⊗ σ ). We have E ∗ ρ,σ = Ω − F ρ ⊗ σ ( F − ∗ )Ω = ρ ⊗ σ ( F Ω − F ∗ F − ∗ Ω) = ρ ⊗ σ ( F ) = E − ρ,σ . (cid:3) We next note that while at the algebraic level, the element Ω defining a unitary involutionof a weak quasi-Hopf C*-algebra may be non-unique, passing to another such operator givesrise to a unitarily equivalent tensor C ∗ -category. Proposition 9.8.
Let A be a weak quasi-bialgebra endowed with the structure of a C*-algebra (or a discrete weak quasi-bialgebra with positive involution). Let Ω and Ω ′ definetwo unitary Ω -involutive structures. Let us upgrade the category of C*-representationsof A to corresponding tensor C ∗ -categories Rep +Ω ( A ) and Rep +Ω ′ ( A ) . Then the functor F : Rep +Ω ( A ) → Rep +Ω ′ ( A ) acting as identity on objects and morphisms admits the structureof a unitary tensor equivalence.Proof. It is easy to check that the functor F becomes a tensor ∗ -equivalence with thenatural transformations F ( ρ ) ⊗ F ( σ ) → F ( ρ ⊗ σ ) acting as identity. The unitary part of thepolar decomposition equips F with the structure of a unitary tensor equivalence by Prop.2.17 b). (cid:3) We next discuss classification of ∗ -representations for the important class of discreteΩ-involutive weak quasi-bialgebras in the sense of Def. 8.5. So we may write, up to ∗ -isomorphism, A = L r M n r ( C ). The projections ρ r : A → M n r ( C ) are irreducible C*-representations. Proposition 9.9.
The ∗ -representations ρ r together with their opposites ρ − r , exhaustthe irreducible ∗ –representations of A up to unitary equivalence. Furthermore any ∗ -representation of A decomposes as an orthogonal direct sum of copies of them. Finally, Ω is positive if and only of for all s , t , ρ s ⊗ ρ t is an orthogonal direct sum of ρ r only.Proof. When we forget about the ∗ -structure, an irreducible representation ρ of A is equiv-alent to some ρ r . Therefore to classify irreducible ∗ –representations, we need to classify upto unitary equivalence the Hermitian forms on C n r making ρ r into a ∗ -representation. By EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 55
Prop. 9.3 these are ρ r and ρ − r . We have already noticed that ρ r and ρ − r are not unitarilyequivalent, hence altogether they form a complete list of irreducible ∗ –representations, upto unitary equivalence. Let now σ be a reducible ∗ -representation of A and let us decom-pose it, as a representation, as a direct sum of certain σ r , where σ r is a multiple of ρ r . Each σ r acts on V σ r = σ ( e r ) V σ , with e r a minimal central projection of A . Hence these subspacesare pairwise orthogonal by ∗ -invariance of σ . In particular, the form of V σ is nondegenerateon all the V σ r . In turn, the pairwise equivalent irreducible summands τ i of a fixed σ r acton the linear span V i of { σ r ( e ) v i , σ r ( e ) v i , . . . , σ r ( e n r ) v i } respectively, where v i form alinear basis of σ r ( e ) V σ r and we claim that it is possible to choose v i pairwise orthogonal.The claim shows that these copies of ρ r act on pairwise orthogonal subspaces. To showthe claim, notice that the map v ∈ σ r ( e ) V σ r → σ r ( e i ) v ∈ σ r ( e ii ) V σ r is unitary betweenpairwise orthogonal subspaces of V σ r , hence the form of V σ r must be nondegenerate oneach of them, and the claim follows. To show the last assertion, we use an orthogonaldecomposition into irreducibles in the general case, given by isometries S ± r,j ∈ ( ρ ± r , ρ s ⊗ ρ t ).These determine the components ρ s ⊗ ρ t (Ω) in the full matrix C*-subalgebras of A ⊗ A by the formula ( ξ, ρ s ⊗ ρ t (Ω) η ) p = P ( S ± r,j ∗ ξ, S ± r,j ∗ η ), where ξ , η vary in the vector space of ρ s ⊗ ρ t and the inner products at the right hand side refer to ρ ± r . The claim easily followsfrom this equation. (cid:3) Remark 9.10.
Examples have been found by Fr¨ohlich and Kerler [46] and Rowell [110,111, 112] of braided fusion categories which are not unitarizable.10.
Unitary braided symmetry and involutive Tannaka-Krein duality
In this section we discuss properties of the involutive structure in a weak quasi-Hopfalgebra concerning the twisting operation, quasitriangular structure and Tannaka-Kreinduality. We start with categorical interpretation of a twist of the ∗ -structure of a weakquasi bialgebra, in analogy with Prop. 5.2 for a twisted bialgebra structure.Let A be a (discrete) weak quasi bialgebra and ( ∗ , Ω) an Ω-involution in the sense ofDef. 8.1. Let t ∈ A (or t ∈ M ( A ) if A is discrete) be a selfadjoint twist, and considerthe corresponding twisted involution ( † , Ω t ), see Prop. 8.14. We thus have two structures( A, ε, ∆ , Φ , ∗ , Ω) and (
A, ε, ∆ , Φ , † , Ω t ) which differ only for their involution. For brevity, wedenote them respectively as A and A t , in analogy with a twist of the bialgebra structure.Consider the functor E : Rep h ( A ) → Rep h ( A t ) defined as follows. If ρ is a ∗ -representationof A then we modify the Hermitian form ( ξ, η ) V ρ of V ρ as ( ξ, η ) t := ( ξ, ρ ( t ) η ) V ρ , andconsider the representation ρ t of A t on the Hermitian space V ρ t so obtained acting as ρ .By construction, ρ t is a † -representation of A t . Proposition 10.1.
Let A be a unitary (discrete) weak quasi bialgebra and t a positivetwist for the involutive structure. Then the functor E : Rep + ( A ) → Rep + ( A t ) taking ρ to ρ t , acting identically on morphisms and with identity natural transformations is a unitarytensor equivalence of tensor C ∗ -categories. Proof.
Pick ρ , σ ∈ Rep + ( A ). For any linear map T : V ρ → V σ , the adjoint of T with respectto the original and modified Hermitian forms are related by T † = σ ( t − ) T ∗ ρ ( t ). Thus if T ∈ ( ρ, σ ) then T † = T ∗ , and this shows that E is a ∗ -functor, which is clearly full, faithfuland essentially surjective, hence a ∗ -equivalence. On the other hand, the tensor structuresof Rep + ( A ) and Rep + ( A t ) are identical, hence E is a tensor equivalence under the identitynatural transformations. To show unitarity we are left to verify that the inner products of ρ t ⊗ σ t and ( ρ ⊗ σ ) t coincide, but this follows from a straightforward computation. (cid:3) It is well known that if A is a quasitriangular quasi-Hopf algebra with R -matrix R , thecategory Rep( A ) has a braided symmetry ε , where the action of ε ( ρ, σ ) on the representa-tion space V ρ ⊗ V σ is given by Σ R , with Σ : V ρ ⊗ V σ → V σ ⊗ V ρ the permutation operator.This construction extends to the weak case. Similarly, if A has an Ω-involution, Rep h ( A )is a braided tensor category as well. We next observe a condition on R assuring unitarityof ε in Rep h ( A ). Proposition 10.2.
Let A be an Ω -involutive weak quasi-bialgebra with quasitriangularstructure defined by R and satisfying ˜ R = R Ω . Then the associated braided symmetry of Rep h ( A ) is unitary. If A is discrete the converse holds.Proof. Our assumption on the R -matrix means R ∗− = Ω R Ω − . The relation betweenthe adjoint morphism ε ( ρ, σ ) ∗ with respect to the ∗ -structure of Rep h ( A ) and the adjoint ε ( ρ, σ ) † with respect to the product form is ε ( ρ, σ ) ∗ = Ω − ε ( ρ, σ ) † Ω. Therefore ε ( ρ, σ ) ∗ = Ω − (Σ ρ ⊗ σ ( R )) † Ω = Ω − ( ρ ⊗ σ ( R ∗ ))Ω Σ = R − Σ = ε ( ρ, σ ) − . (cid:3) Remark 10.3.
The assumptions in Prop. 10.2 may be read as saying that the twistrelation ˜ A = A Ω holds not only at the level of weak quasi-bialgebras, but also for theirnatural quasitriangular structures, cf. Prop. 7.2. Furthermore if ˜ R = R Ω holds for a givenΩ-involutive quasitriangular weak quasi-bialgebra with R -matrix R and involution Ω thenthey hold for any twisted algebra with twisted R -matrix R T and twisted involution Ω T ,Ω − T as defined in c) of Prop. 7.2 and Prop. 8.8 respectively. Corollary 10.4.
Let A be a finite dimensional discrete weak quasi-Hopf algebra with aquasitriangular structure R . Then any involution ( ∗ , Ω) making A into a unitary weakquasi bialgebra satisfies ˜ R = R Ω .Proof. The tensor C ∗ -category Rep + ( A ) is braided and fusion, hence by Theorem 3.2 in[49] the braided symmetry is unitary. We may then apply Prop. 10.2. (cid:3) We next discuss a version of Tannaka-Krein duality for Ω-involutive weak quasi bialge-bras. Recall that unitarity of a weak quasi tensor ∗ -functor was defined in Def. 2.14, andthat triviality of an Ω-involution is introduced in Def. 8.9. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 57
Theorem 10.5.
Let C be a semisimple tensor ∗ -category, with finite dimensional morphismspaces F : C → Herm a faithful weak quasi tensor ∗ -functor defined by ( F, G ) and A =Nat ( F ) be the discrete weak quasi bialgebra associated to F as in Th. 5.6 and Th. 6.10endowed with its natural involution ∗ . Then a) the element Ω ∈ A ⊗ A defined by Ω ρ,σ = F ∗ ρ,σ ◦ F ρ,σ makes A into an Ω -involutiveweak quasi bialgebra, b) there is a canonical unitary tensor ∗ -functor E : C → Rep h ( A ) and is an equivalence.Furthermore, the composite of E with the forgetful functor Rep h ( A ) → Herm isunitarily monoidally isomorphic to F , c) ( F , F, G ) is (strongly) unitary if and only if A the Ω -involution of A as in a) is(strongly) trivial, d) when C is unitary and F : C → Hilb then A is a unitary weak quasi-bialgebra and E is a unitary tensor equivalence between E : C → Rep + ( A ) .Proof. a) For simplicity in the following computations we drop the indices of the naturaltransformations. Note that Ω is selfadjoint, and in particular positive when F takes valuesin Hilb. Furtherore, Ω has ∆( I ) = GF as domain and ∆( I ) ∗ = ( GF ) ∗ as range. Weset Ω − := GG ∗ . We have: Ω − Ω = GG ∗ F ∗ F = G ( F G ) ∗ F = GF = ∆( I ) and similarlyΩΩ − = ∆( I ) ∗ . Furthermore, for η ∈ A ,Ω∆( η ∗ ) = F ∗ F ∆( η ∗ ) = F ∗ F Gη ∗ ρ ⊗ σ F = F ∗ η ∗ ρ ⊗ σ F = F ∗ η ∗ ρ ⊗ σ G ∗ F ∗ F = ∆( η ) ∗ Ω . We have thus verified axioms (8.1), (8.2), (8.3), while (8.4) follows easily from (2.2) and(8.5) can be checked with computations similar to those above. b) By assumption, F ( ρ ) isan Hermitian space and by Theorem 5.6 E ( ρ ) is a representation of A on F ( ρ ) and E is atensor equivalence with Rep( A ) and therefore also with Rep h ( A ). It is easy to check that E is ∗ -preserving, it follows that E takes values in Rep h ( A ). To show unitarity of E recallthat the tensor structure of E regarded as a morphism in Rep h ( A ) is F ρ,σ . We computethe adjoint F ∗ ρ,σ in Rep h ( A ). As before, we momentarily denote by † the usual adjoint ofthe tensor category of Hilbert spaces. We have F ∗ ρ,σ = Ω − ρ,σ F † ρ,σ = G ρ,σ G † ρ,σ F † ρ,σ = G ρ,σ ( F ρ,σ G ρ,σ ) † = G ρ,σ . c) By definition of unitarity of ( F , F, G ), F ∗ F = P ∗ P and GG ∗ = P P ∗ , with P = GF andthis by construction corresponds to triviality of the Ω-involution of A , and similarly for therelation between strong unitarity of the weak quasi-tensor structure and strong trivialityof the Ω-involution. (cid:3) Remark 10.6.
Theorem 10.5 for unitary weak quasi-bialgebras has origin in [59] wherethe author assumes that F ∗ ρ,σ = G ρ,σ and are isometries, that is a strongly unitary structurein our terminology. In this case he proves that the ∗ -involution of A commutes with thecoproduct. We note that the examples that we discuss in Sect. 24 arising from quantumgroups at roots of unity do not satisfy this property, and this motivated us to consider themore general case. Example 10.7.
Consider the ∗ -category C = Herm ωG of G -graded Hermitian spaces. Itbecomes a tensor ∗ -category with natural tensor product and associator given by a T -valued3-cocycle ω over G . For every g ∈ G , denote by C + g ( C − g ) the one-dimensional Hermitianspace of degree g with positive (negative) scalar product. Then C + g and C − g are twoirreducible equivalent but not unitarily equivalent objects of Herm ωG , and C ± g and C ± h areinequivalent for g = h . The category Herm ωG contains Hilb ωG as a full tensor C ∗ -subcategorywith restricted ∗ -structure. Consider F : Herm ωG → Herm the forgetful functor. Note that F preserves the Hermitian forms, thus it takes a definite sign on the unitarily inequivalentsimple objects. It follows that Nat ( F ) is a pre- C ∗ -algebra that may be identified withthe C ∗ -algebra of complex-valued functions on G . Note that F ( g ) ⊗ F ( h ) and F ( gh ) areunitarily equivalent Hermitian spaces with definite forms, thus every quasitensor structure F g,h on F satisfies Ω( g, h ) := F ∗ g,h F g,h >
0. It follows from Theorem 10.5 that A = Nat ( F )is a unitary pointed quasi-bialgebra which identifies with C ω ( G ) with unitary structuredefined by Ω as in Example 8.6. Note that by the last part of Example 8.6 there existexamples of pointed tensor ∗ -categories which are not unitarily equivalent to some Herm ωG . Remark 10.8.
In Sect. 8 we have constructed the unitarization A Ω / associated to aunitary discrete weak quasi-bialgebra A in the case where (Ω − ) / is a left inverse ofΩ / . This construction may be described categorically as follows. Let ( F , F , G ) be afaithful weak quasi tensor ∗ -functor of a semisimple unitary tensor category C and A theassociated unitary discrete weak quasi bialgebra with involution denoted ( ∗ , Ω) followingTheorem 10.5. If this functor is non-unitary and for example we know that satisfies theleft inverse property (2.8) then we may consider the unitarized functor ( F , F ′ , G ′ ) as inpart a) of Prop. 2.17, see also Def. 2.18. This new structure in turn gives rise to a newunitary weak quasi bialgebra B corresponding to the unitarization A Ω / of A , by the proofof Theorem 5.9 with trivial unitary structure by Cor. 8.12. This structure is also stronglytrivial by Prop. 2.16.The notion of unitarization will have a useful extension in Sect. 23 in that will be appliedto more useful situations in subsequent sections.We ask how to construct and parameterise faithful ∗ -functors G : C → Hilb from a C*-category. If G is given, we may construct new ∗ -functors to Hilb via a categorical analogueof the twist deformation of the involution of an algebra of Prop. 8.14 in the followingway. Let t ∈ Nat( G ) be a positive invertible natural transformation and let G t ( ρ ) be G ( ρ )as a vector space, but with modified inner product ( ξ, η ) t := ( ξ, t ρ η ) G ( ρ ) . The action of G t on morphisms is the same as that of G . The fact that G is ∗ -preserving together withnaturality of t easily imply that G t is ∗ -preserving as well, hence a ∗ -functor. The ∗ -algebras A = Nat ( G ) and B = Nat ( G t ) are related by B = A t .Faithful functors F : C → Vec are described, up to isomorphism, by functions D :Irr( C ) → N thanks to Theorem 5.9 (a). We thus need to parameterize the ways how F canbe written as F = HG with G : C → Hilb is a ∗ -functor and H : Hilb → Vecthe forgetful functor.
EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 59
Proposition 10.9.
Let C be a C*-category with finite dimensional morphism spaces and F : C → Vec a faithful functor. Then F factors through F = HG where G : C → Hilb is afaithful ∗ -functor and H : Hilb → Vec the forgetful functor. Any other ∗ -functor G ′ withthe same properties is of the form G t for a unique positive invertible t ∈ Nat( G ) .Proof. We choose, for each ρ ∈ Irr( C ), a positive inner product on F ( ρ ), and let G ( ρ )the corresponding Hilbert space. Note that F ( T ∗ ) = F ( T ) ∗ holds for T ∈ ( ρ, ρ ) for anychoice of inner product when ρ is irreducibile, since these morphisms are scalars and F is linear. We use orthogonal complete reducibility of µ ∈ C via isometries S ρ,i ∈ ( ρ, µ )with ρ irreducibles, to extend the construction of a Hilbert space G ( µ ) to all objects µ via( ξ, η ) G ( µ ) := P ρ,i ( F ( S ∗ ρ,i ) ξ, F ( S ∗ ρ,i ) η ) G ( ρ ) . It follows that the inner product is independentof the choice of the isometries S ρ,i . Letting G act as F on morphisms, one sees that G ( S ∗ ρ,i ) = G ( S ρ,i ) ∗ and this implies G is ∗ -preserving. Another decomposition F = HG ′ gives a new Hilbert space structure G ′ ( ρ ) on the same vector space as G ( ρ ), hence we mayfind a unique positive invertible operator t ρ on G ( ρ ) such that ( ξ, η ) G ′ ( ρ ) = ( ξ, t ρ η ) G ( ρ ) . Since G ′ is a ∗ -functor, this implies that t ∈ Nat( G ). (cid:3) We summarise the main results of this and previous sections.
Corollary 10.10.
Let C be a tensor C ∗ -category with finite dimensional morphism spacesand D a weak dimension function on C . Then there is a faithful weak quasi-tensor ∗ -functor G : C → Hilb such that D ( ρ ) = dim( G ( ρ )) . If A is the discrete unitary weak quasibialgebra corresponding to G via duality then all the others corresponding to different weakquasi-tensor ∗ -functors with the same dimension function are isomorphic to A F,t for sometwist F ∈ A ⊗ A and t ∈ A + of the bialgebra and Ω -involutive structure of A respectively. Unitarizability of representations and rigidity
In order to construct objects of Rep h ( A ), or establish rigidity of that category, we needto know which representations of A are equivalent to ∗ –representations on non-degenerateHermitian spaces. Recall from (3) the notion of conjugate object in a tensor ∗ -category.Thus a ∗ –representation ρ ∈ Rep h ( A ) has a conjugate in Rep h ( A ) if and only if the canon-ical right dual ρ c introduced in Def. 5.3 is equivalent to a ∗ -representation, the conjugate ρ . If this is the case the canonical left dual c ρ will be automatically equivalent to ρ c and ρ as well, and the double dual ρ cc to ρ .If a weak quasi-Hopf algebra A has an involution ∗ : A → A , to any finite dimensionalrepresentation ρ on a vector space we may associate two more representations, ρ c and c ρ both acting on the conjugate vector space V ρ via ρ c ( a ) ξ = ρ ( S ( a ) ∗ ) ξ, c ρ ( a ) ξ = ρ ( S − ( a ) ∗ ) ξ. (Alternatively, we may consider the representations acting on V ρ via ξ → ρ ( S ( a ∗ )) ξ and ξ → ρ ( S − ( a ∗ )) ξ , respectively equivalent to c ρ and ρ c by Prop. 8.16.)If the involution makes A into an Ω-involutive a weak quasi-Hopf algebra, then thereare equivalences ρ cc ≃ ρ ≃ cc ρ thanks to Prop. 8.16 again. Proposition 11.1.
Let A be an Ω -involutive weak quasi-Hopf algebra, and ρ a finite di-mensional vector space representation of A . a) If ρ is equivalent to a ∗ -representation then there is an equivalence Φ : ρ c → ρ c (resp. Φ ′ : c ρ → c ρ ) related to the Hermitian form of ρ via ( ξ, η ) = Φ ξ ( η ) , b) if ρ is irreducible and if ρ c ≃ ρ c (or c ρ ≃ c ρ ) then ρ is equivalent to a ∗ -representationand the associated Hermitian form is unique up to a nonzero real scalar.Proof. a) If ρ is equivalent to the ∗ -representation σ via the invertible T ∈ ( ρ, σ ) we mayendow the space of ρ with the nondegenerate Hermitian form making T unitary, and inthis way ρ becomes a ∗ -representation. It follows that we may canonically identify theconjugate space V ρ with V ∗ ρ with via the invertible map ξ → Φ ξ , which is the functional η → ( ξ, η ). A computation shows that this makes ρ c ( a ) equivalent to the representationacting on ξ ∈ V ρ as ρ ( S ( a )) ∗ ξ = ρ ( S ( a ) ∗ ) ξ = ρ c ( a ) ξ . (Similarly, c ρ turns into c ρ .) b)Let Φ ∈ ( ρ c , ρ c ) be an invertible morphism, and introduce a sesquilinear form on V ρ by( ξ, η ) = Φ ξ ( η ), clearly nondegenerate. Let us define the right and left adjoint of a linearmap T : V ρ → V ρ respectively by ( T ∗ ξ, η ) = ( ξ, T η ) and ( ξ, ∗ T η ) = (
T ξ, η ). A computationusing the intertwining property of Φ shows that for a ∈ A , ρ ( a ) ∗ = ρ ( a ∗ ) = ∗ ρ ( a ). Let usintroduce an inner product ( ξ, η ) pos in V ρ making some basis orthonormal, let T → T † bethe corresponding adjoint map and B : V ρ → V ρ be the unique invertible map such that( ξ, η ) = ( ξ, Bη ) pos . Then T ∗ = B − † T † B † and ∗ T = B − T † B . Equating ρ ( a ) ∗ = ∗ ρ ( a )gives B − † Bρ ( a ) = ρ ( a ) B − † B , hence B † is a scalar multiple of B by irreducibility of ρ . But k B k = k B † k (norm associated to ( ξ, η ) pos ) and it follows that this scalar lies in T . Hence after rescaling B we get B † = B , and finally derive that ( ξ, η ) is Hermitian.Finally, with a similar argument, if ρ is irreducible and unitary on a Hermitian space withHermitian form ( ξ, η ) then any other nondegenerate Hermitian form on the same spacemaking ρ ∗ -invariant, when written as ( ξ, Aη ) with A invertible and selfadjoint, impliesthat A is a real scalar. (cid:3) Corollary 11.2.
Let A be a discrete Ω -involutive (unitary) weak quasi-Hopf algebra. Thenevery representation is equivalent to a ∗ -representation ( C ∗ -representation). In particular Rep h ( A ) ( Rep + ( A ) ) is rigid and the forgetful functor Rep h ( A ) → Rep( A ) ( Rep + ( A ) → Rep( A ) ) is a tensor equivalence.Proof. Let ρ be a representation of A that we may assume irreducible by complete re-ducibility. Note that the antipode S permutes the minimal central idempotents of A andthat these idempotents are selfadjoint since the involution of A is positive by assumption,see Def. 8.5. This implies that ρ c and ρ c have the same central support, and therefore theyare related by an isomorphism T . We may then apply Prop. 11.1. Note also that a nonzeroscalar multiple of T induces a positive inner product on the space of ρ by the classificationof Hermitian forms associated to involutive discrete weak quasi-Hopf algebras, Prop. 9.9,hence the conclusion follows also in the case where Ω is positive. (cid:3) By the end of the section we shall identify the conjugates in the tensor C ∗ -categoryRep + ( A ) in the discrete w -Hopf case. We next discuss some results guaranteeing rigidity EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 61 in possibly non-semisimple tensor categories motivated by the work of Kashiwara, Kirillov,Wenzl for U q ( g ) at roots of unity [72, 79, 128]. Recall that the element ω was defined inProp. 8.16. Proposition 11.3.
Let A be an Ω -involutive weak quasi-Hopf algebra and ρ a ∗ -representationequivalent to ρ cc . Then a) ρ c is equivalent to a ∗ –representation if and only if there is an invertible K ρ ∈ ( ρ, ρ cc ) such that F ρ := K ρ ρ ( ω ∗ ) is selfadjoint with respect to the Hermitian form of ρ . Inthis case, the forms making ρ c into a ∗ -representation are parametrised by K ρ via ( ξ, η ) = ( η, F ρ ξ ) , b) if ρ is a C*-representation then ρ c is equivalent to a C*-representation if and onlyif F ρ can be chosen positive, c) if ρ is irreducible then ρ c is equivalent to a ∗ -representation. The associated K ρ ∈ ( ρ, ρ cc ) is unique up to a real scalar multiple.Proof. a) If ρ c is equivalent to a ∗ –representation then ρ cc and ρ cc are equivalent by theprevious proposition, and let Ψ be this equivalence. We may write Ψ as the composite of ρ ( ω ∗ ) : ρ cc → ρ with an equivalence K ρ : ρ → ρ cc in turned followed by Φ t : ρ cc → ( ρ c ) c ,where Φ : ρ c → ρ c is defined as in the proof of a) of Prop. 11.1, and Φ t is the transposedof Φ. The Hermitian form making ρ c into a ∗ -representation is given by ( ξ, η ) = Ψ ξ ( η ).An explicit computation shows that this is precisely the form in the statement. Con-versely, for any K ρ ∈ ( ρ, ρ cc ), the sesquilinear form defined by F ρ = K ρ ρ ( ω ∗ ) is Her-mitian (positive) precisely when F ρ is selfadjoint (positive). A computation shows that( ρ c ( a ) ξ, η ) = ( ξ, ρ c ( a ∗ ) η ), in other words ρ c is a ∗ -representation. The proof of b) is nowclear. c) By irreducibility and b) of Prop. 11.1, it suffices to show that ρ cc ≃ ( ρ c ) c . Nowsuch an equivalence can be obtained as in the proof of a) starting from the choice of aninvertible K ρ ∈ ( ρ, ρ cc ). (cid:3) Corollary 11.4.
Let A be an Ω -involutive weak quasi-Hopf algebra with an antipode ( S, α, β ) such that S commutes with ∗ . Then a) every ∗ -representation ρ has ρ c as a conjugate in Rep h ( A ) with respect to the formconjugate to that of ρ : ( ξ, η ) = ( η, ξ ) . Hence Rep h ( A ) is rigid. b) If ρ is a C*-representation, so is ρ c . Hence if A is a unitary weak quasi-Hopfalgebra, Rep + ( A ) is rigid as well.Proof. We may take K ρ = ρ ( ω ∗− ) by Prop. 8.17, hence F ρ = I for all ∗ -representations ρ . (cid:3) Remark 11.5.
Let A = Nat ( F ) be the discrete weak quasi-bialgebra associated to asemisimple tensor category C endowed a weak quasi-tensor functor F : C → Vec as inTheorem 5.6. When C is also a C*-category and F factors through a ∗ -functor F : C → Hilbthen A has a natural pre-C*-algebra involution. If C is rigid and the dimension assumptionof Prop. 5.6 (d) hold (e.g. C is a fusion category) then A has an antipode ( S, α, β ). Wenote that S may always be chosen commuting with ∗ . Indeed, following the proof of Theorem 5.6 (d), for each ρ , F ( ρ ) ∗ identifies naturally with the conjugate vector space F ( ρ ), which we endow with the unique Hilbert space structure making the conjugationmap J : F ( ρ ) → F ( ρ ) antiunitary. It also follows that a transposed linear map L t identifieswith J L ∗ J − . On the other hand, we may choose the natural transformation U unitary.It follows from the antipode formula given in the proof that S ( η ∗ ) = S ( η ) ∗ . For exampleif C = Rep( A ) with A a discrete weak quasi-Hopf algebra which is also a pre-C*-algebrathen the procedure reconstructs the original antipode of A when this commutes with ∗ byRemark 5.7 c), but it gives a new one otherwise. Example 11.6.
We next describe the conjugate equations in Rep h ( A ) (Rep + ( A )) underthe assumption that S commutes with ∗ . Given a ∗ -representation ρ , we may use thecanonical identification of ρ c with ρ c and obtain from Prop. 5.4 the following solution forthe pair ρ , ρ c ∈ Rep h ( A ), r ρ = d ∗ ρ = Ω − n X i =1 µ i e i ⊗ α ∗ e i , r ρ = b ρ = n X i =1 βe i ⊗ µ i e i (11.1)with e i a basis of the space of ρ satisfying ( e i , e j ) = δ i,j µ i and µ i = ±
1. Let us considerthe case of Rep + ( A ), so µ i = 1. Then it follows by a straightforward computation that r ρ = Ω − P i e i ⊗ α ∗ e i = P i e i ⊗ ( α Ω ) ∗ e i , r ∗ ρ ξ ⊗ η = ( β Ω η, ξ ) and this implies r ∗ ρ r ρ = d ρ r ρ = Tr( α ( α Ω ) ∗ ) , r ∗ ρ r ρ = r ∗ ρ b ρ = Tr(( β Ω ) ∗ β ) , (11.2)where α Ω and β Ω are defined in (4.11). When Ω = ∆( I ) is trivial then a computation showsthat α Ω = α and β Ω = β . If α , β are in addition unitary then the intrinsic dimensionscoincide with the vector space dimensions. In Sect. 24 we shall discuss examples of Ω-involutive weak quasi-Hopf algebras A = A ( g , q, ℓ ) arising from a certain semisimplifiedquotient category associated to quantum groups at roots of unity U q ( g ). In this case theantipode is of Kac type but Ω is non-trivial, compatibly with non-integrality of intrinsicdimensions.We next construct a natural solution of the conjugate equations for objects of Rep + ( A ),with A a unitary discrete w-Hopf algebra not necessarily of Kac type. Our methods extendthose of [131, 36, 124] for the case of discrete or compact quantum groups. We first establishexistence of a Haar element. Proposition 11.7.
Let A be a discrete Ω -involutive weak quasi-bialgebra. There is aunique nonzero selfadjoint idempotent h ∈ A such that ah = ha = ε ( a ) h for all a ∈ A .Proof. The proof is as in Prop. 3.1 in [124]. The counit is an irreducible ∗ -representationof A . As such, it coincides with the projection onto one of its one dimensional matrixsubalgebras. The idempotent defining this component is the desired element h . (cid:3) Definition 11.8.
The element h is called the Haar element .The following lemma extends a known idea in the framework of coassociative quantumgroups which, to our knowledge, dates back to [131]. Here we consider a modification dueto non-triviality of the associator, where the need of the special form that the associator
EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 63 takes for w-Hopf algebras is apparent. We are not aware of validity of an analogous lemmain a general quasi-coassociative framework.
Lemma 11.9.
Let A be a discrete w-Hopf algebra. Then ∆( I ) A ⊗ A = ∆( A ) I ⊗ A, A ⊗ A ∆( I ) = A ⊗ I ∆( A ) . Proof.
We write ∆( I ) = a ⊗ b and for a generic x ∈ A , ∆( x ) = x ⊗ x . Consider the linearmap T : A ⊗ A → A ⊗ A defined by T ( x ⊗ y ) = xy ⊗ y . We show that T is surjective,and this gives the second stated relation. It is straightforward to see that T coincides withthe map ˜ T : A ⊗ A → A ⊗ A defined by ˜ T ( x ⊗ y ) = xS ( a ) a y ⊗ by . Consider also themap R given by R ( x ⊗ y ) = xS ( y ) ⊗ y . We have T R ( x ⊗ y ) = T ( xS ( y ) ⊗ y ) = ˜ T ( xS ( y ) ⊗ y ) = xS ( a y ) a y , ⊗ by , . We use the associativity relation a y ⊗ a y , ⊗ by , = y , a ⊗ y , b ⊗ y b and get aftera brief computation T ( xS ( y ) ⊗ y ) = [ x ⊗ y ][ S ( a ) b ⊗ b ]. A slight modification of thisidea gives T ( xS ( y ) ⊗ y b ′ ) = ˜ T ( xS ( y ) ⊗ y b ′ ) = [ x ⊗ y ][ S ( a ) b b ′ ⊗ yb b ′ ]. We replace x by ˜ x = xS ( a ′ ) and y by ˜ y = yS ( a ′ ) and obtain T R (˜ x ⊗ ˜ y ) = [ x ⊗ y ] f , where theelement f was defined in Prop. 4.12 for general weak quasi-Hopf algebras and consideredagain in Prop. 7.8 for w-Hopf algebras. Since f is partially invertible with domain ∆( I ),the proof is complete. The first relation can be proved in a similar way with the maps T ′ ( x ⊗ y ) = x ⊗ x y as R ′ ( x ⊗ y ) = x ⊗ S ( x ) y . (cid:3) The following relations extend Prop. 4.1 of [124] to our setting.
Proposition 11.10.
Let A be a discrete w-Hopf algebra. For all x , y ∈ A we have ∆( h ) x ⊗ y = ∆( h ) I ⊗ S ( x ) y, x ⊗ y ∆( h ) = xS ( y ) ⊗ I ∆( h ) . Proof.
We only show the first relation. We write ∆( I ) x ⊗ y as a finite sum of elements ofthe form ∆( p ) I ⊗ q , thanks to the first relation of Lemma 11.9. Evaluating m ◦ S ⊗ S ( x ) y = ε ( p ) q . On the other hand∆( h ) x ⊗ y = ∆( hp ) I ⊗ q = ∆( hε ( p )) I ⊗ q = ∆( h ) I ⊗ ε ( p ) q = ∆( h ) I ⊗ S ( x ) y, and the relation follows. (cid:3) The following result gives a canonical implementing element for the squared antipode.We omit the proof as it equals that of Prop. 4.3 in [124]. For every full matrix subalgebra M r ( C ) we let e r denote its identity, regarded as a central projection of A , Tr r the tracemap which takes value 1 on the minimal idempotents, and r ′ the unique index such that S ( M r ( C )) = M r ′ ( C ), which is the same as S ( M r ′ ( C )) = M r ( C ). Proposition 11.11.
Let A be a discrete w-Hopf algebra. Then S ( x ) = KxK − for all x ∈ A , where K = ( K r ) ∈ M ( A ) is given by K r = [Tr r ′ ⊗ h ))] − ∈ M r ( C ) . Theorem 11.12.
Let A be an unitary discrete w-Hopf algebra. Then for every C*-representation ρ , the invertible operator F ρ := ρ ( Kω ∗ ) is positive. Therefore ρ c becomes aconjugate of ρ in Rep + ( A ) with inner product ( ξ, η ) = ( η, F ρ ξ ) .Proof. It suffices to show positivity of F ρ for the C*-representations ρ r which project ontothe matrix algebras M r ( C ), since any other ρ is unitarily equivalent to a direct sum ofthem. We note that ∆( h )Ω − is positive in M ( A ⊗ A ), as∆( h )Ω − = ∆( h ) Ω − = ∆( h )Ω − ∆( h ) ∗ . Hence ∆( h )Ω − e r ′ ⊗ e r is positive as well. Using the notation Ω − = x ⊗ y , we have, thanksto Prop. 11.10,∆( h )Ω − e r ′ ⊗ e r = ∆( h ) I ⊗ S ( xe r ′ ) ye r = ∆( h ) I ⊗ S ( x ) ye r = ∆( h ) I ⊗ ω r , with ω r the component of ω along M r ( C ). Evaluating the positive map Tr r ′ ⊗ K − r ω r = ρ r ( K − ω ) is positive. Hence ρ r ( Kω ∗ ) = ω r ρ r ( K − ω ) − ω ∗ r ispositive as well. (cid:3) Turning C ∗ -categories into tensor C ∗ -categories, I The problem of constructing unitary tensor categories is of great importance in connec-tion with the study of fusion categories from quantum groups at roots of unity or conformalfield theory. In the former setting, a natural ∗ -structure was introduced by Kirillov [79]for certain even roots of unity, and unitarity was shown by Wenzl and Xu [128, 133]. Atensor category is called unitarizable if it is tensor equivalent to a tensor C*-category. Wehave observed in 9.10 that examples of non-unitarizable fusion categories from quantumgroups and certain roots of unity are known.We start with the following setting, which will be called condition a).a) Let C be a tensor category and C + a C*-category, and assume that we have anequivalence of linear categories F : C + → C . We shall always assume that every object of C + is completely reducible into a finite directsum of irreducibles. We wish to upgrade C + to a tensor C ∗ -category via F .In this section we discuss a result which characterizes when a solution exists and isunique. We shall derive two variants, the first applies to unitarizable tensor categoriesand will be useful in Sect.17 where we shall construct unitary tensor structures for theC*-category of unitary representations of several classes of Vertex Operator Algebras. Themain strategy is that of constructing unitary tensor structures on tensor categories ortransfer them from old structures to new structures. We also note that this result givesa positive answer to a question posed by Galindo in [49] on uniqueness of unitary tensorstructures on tensor categories. The second variant will be useful in Sect. 21, 23, 24where unitary weak quasi-Hopf algebras will be constructed with a direct method from thebraiding for certain general ribbon categories and in particular for those arising from thequantum groups at roots of unity. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 65
Definition 12.1.
Let F : C + → C satisfy a). We shall say that the tensor structure of C is transportable compatibly with the C*-structure , or simply C*-transportable to C + if C + can be upgraded to a tensor C ∗ -category in such a way that F : C + → C becomes a tensorequivalence.We note that C*-transportability will be possible only in certain circumstances. Forexample, if C is a finite semisimple tensor category then we know that C is tensor equivalentto some Rep( A ), with A a semisimple weak quasi bialgebra. Since A admits the structureof a C ∗ -algebra, the C ∗ -category C + of C ∗ -representations of A satisfies a). On the otherhand, if C is not tensor equivalent to a tensor C ∗ -category, see Remark 9.10, then C + doesnot admit any tensor C ∗ -structure that makes it tensor equivalent to C .We shall describe two main classes of tensor categories for which tensor structure aretransportable compatibly with the C*-structure, and two upgrading of C + correspondingto a C*-transportable tensor structure of C provide unitarily tensor equivalent tensor C ∗ -categories. The notion of weak quasi Hopf algebra will play a prominent role.In the mentioned application, C plays the role of a category of infinite dimensional rep-resentations of interest of some algebraic structure endowed with a ‘fusion’ tensor product,and C + the category of unitary representations on Hilbert spaces. The functor F : C + → C is understood as that which forgets the unitary structure. The assumption that it be anequivalence means that every object of C can be made into a unitary representation, anassumption which is known to hold in a variety of circumstances as clarified in the lastsection. Or else C may be taken as Andersen fusion category of a quantum group U q ( g ) atroots of unity for the values q = e iπ/ℓd . In this case, a first part of Wenzl theory consistsin showing indeed that C is a C*-category in a natural way. We thus see from these twoexamples that the problem in our formulation includes that of unitarizing representationcategories of VOAs and also a substantial part of Wenzl-Xu theory.A note on notation. Since we shall deal at the same time with semisimple linear or C*or tensor categories, and sometimes we shall use only part of the structure, for a quickexplanation of the available or involved structure, we shall use a suffix + on a category todenote that it is a C*-category and on a functor if it is ∗ -preserving. Continuous arrowsdenote tensor equivalences, and dashed arrows linear equivalences. Thus a commutativediagram where only part of the categories or equivalence is tensorial, are understood atthe level of functors. Definition 12.2.
Let F : C + → C be as in a). Let A be a discrete weak quasi bialgebra en-dowed with an involution of pre-C*-algebra, and consider, accordingly, the tensor categoryRep( A ) and the C ∗ -category Rep + ( A ). A triple ( A, E + , E ) constituted by a ∗ -equivalence E + : C + → Rep + ( A ) and a tensor equivalence E : C → Rep( A ) will be called compatible with F if the following diagram commutes C + Rep + ( A ) C Rep( A ) E + F F A E where F A : Rep + ( A ) → Rep( A ) is the forgetful functor.A compatible triple defines a weak dimension function on C via D ( ρ ) := dim( E ′ ( ρ )),where E ′ is the composite of E with the forgetful functor Rep( A ) → Vec. We next see thatcompatible triples may be constructed and classified under mild assumptions.
Proposition 12.3. If ( A, E , E + ) is a compatible triple for F : C + → C then for any twist F ∈ A ⊗ A of the weak quasi bialgebra structure and any positive twist t ∈ A of the ∗ -involution, the twisted algebra A F,t is part of another compatible triple with the same weakdimension function and they are all of this form.Proof.
The proof follows from Prop. 5.2, Theorem 5.9 and part of Prop. 10.1. (cid:3)
Remark 12.4.
As we shall see, natural constructions in conformal field theory, give riseto canonically associated associative algebras A , the Zhu algebras, and also to linear func-tors E , which are already known to play an important role in the theory of VOAs. Theconstruction of compatible triples for these remarkable examples is our main motivationin the definition. Proposition 12.5. F : C + → C admits a compatible triple if and only if C admits anintegral weak dimension function.Proof. The notion of a compatible triple ( A, E , E + ) may equivalently be given via an ab-stract construction as follows. There is a canonical isomorphism of algebras φ : A → Nat ( E ′ ) which induces an isomorphism of categories φ ∗ : Rep(Nat ( E ′ )) → Rep( A ) suchthat φ ∗ e E ′ = E , where e E ′ : C → Rep(Nat ( E ′ )) is the equivalence arising from Tannaka-Krein reconstruction of E ′ . There is also an isomorphism of ∗ -algebras A → Nat ( E + ′ ).The compatibility condition implies H ( E + ) ′ = E ′ F , with H : Hilb → Vec the forgetfulfunctor. These remarks together with Tannaka-Krein duality results imply that giving acompatible triple is the same thing as giving a faithful ∗ -functor E + ′ : C + → Hilb and afaithful weak quasi-tensor functor E ′ : C → Vec such that E ′ F = HE + ′ . Now it suffices toapply Theorem 5.9, 10.9. (cid:3) Theorem 12.6.
Let F : C + → C satisfy a). Assume that C admits a weak dimensionfunction D , and let ( A, E + , E ) be a compatible triple with dimension D . Then the tensorstructure of C is C*-transportable to C + via F if and only if A can be upgraded to a unitaryweak quasi bialgebra compatible with the given involution on A . If this is the case, thediagram defining the triple becomes a commuting diagram of tensor equivalences and E + can be chosen unitary. Furthermore, any two tensor C ∗ -completions of C + obtained froma C*-transportable F yield unitary tensor equivalent tensor C ∗ -categories.Proof. Following the proof of Prop. 12.5, and adopting the same notation, we shall identify A with Nat ( E + ′ ) as a ∗ -algebra. If C + admits the structure of a tensor C ∗ -categoryover the underlying C*-category such that F : C + → C becomes a tensor equivalencethen the composite of the left with the bottom equivalences in the diagram is a tensorequivalence hence, by commutativity of the diagram, the composite of top with the right EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 67 equivalences C + → Rep + ( A ) → Rep( A ) is a tensor equivalence as well. On the other hand,Rep( A ) → Vec is a weak quasi-tensor functor, hence so is the composite C + → Rep + ( A ) → Rep( A ) → Vec. But this functor factors through C + → Rep + ( A ) → Hilb → Vec andHilb → Vec is both a forgetful functor and a tensor equivalence, and this implies that E + ′ : C + → Rep + ( A ) → Hilb is a ( ∗ -preserving) weak quasi-tensor functor. It follows thatNat ( E + ′ ) can be made into a unitary weak quasi bialgebra and E + into a unitary tensorequivalence by Theorem 10.5. This structure can be transferred to A , and therefore iscompatible with the given ∗ -involution of A . It is now easy to see that it extends the givenweak quasi-bialgebra structure on A .Conversely, if A admits the structure of a unitary weak quasi bialgebra with the given ∗ -structure then, by Corollary 9.5, Rep + ( A ) is a tensor C ∗ -category tensor equivalent toRep( A ) and hence to C . The top equivalence of the diagram defining a compatible tripleacts from the linear category C + to the tensor category Rep + ( A ). It is a general fact thatunder this circumstance, C + can be made into a tensor category in such a way that E + isbecomes a tensor equivalence. Indeed, given objects ρ , σ ∈ C + , we define a tensor productobject ρ ⊗ σ in C + ρ ⊗ σ := S + (cid:0) E + ( ρ ) ⊗ E + ( σ ) (cid:1) , and a tensor product morphism by a similar formula, S ⊗ T := S + (cid:0) E + ( S ) ⊗ E + ( T ) (cid:1) . Here S + : Rep + ( A ) → C + is an inverse equivalence of E + , Moreover, if α denotes the unitaryassociator in Rep + ( A ) we define the unitaries α ′ ρ,σ,τ : ( ρ ⊗ σ ) ⊗ τ → ρ ⊗ ( σ ⊗ τ )by α ′ ρ,σ,τ := S + (1 E + ( ρ ) ⊗ η − E + ( σ ) ⊗ E + ( τ ) ◦ α E + ( ρ ) , E + ( σ ) , E + ( τ ) ◦ η E + ( ρ ) ⊗ E + ( σ ) ⊗ E + ( τ ) ) . where η : E + S + → E + is ∗ -preserving, S + may be chosen ∗ -preserving, η unitary by Prop. 2.11, and Rep + ( A ) is atensor C ∗ -category, it is immediate to check that that the relation ( S ⊗ T ) ∗ = S ∗ ⊗ T ∗ holdson morphisms and α ′ is unitary. This gives the C*- tensor structure on C + . Moreover, E + becomes a tensor equivalence with unitary tensor structure E ρ,σ := η − E + ( ρ ) ⊗ E + ( σ ) . Sincethe forgetful Rep + ( A ) → Rep( A ) is a tensor equivalence as well with the trivial tensorstructure, it follows that F : C + → C has a unique tensor structure such that EF = F A E + as tensor functors.Uniqueness. Let us next consider a new tensor C ∗ -category C ′ coinciding with C + asa C*-category and making F into a new tensor equivalence F ′ . Applying the above con-struction in the opposite direction, that is with S + in place of E + , gives a new tensor C ∗ -category structure to Rep + ( A ), denoted Rep ′ ( A ) and new unitary tensor equivalences S ′ : Rep ′ ( A ) → C ′ and E ′ : C ′ → Rep ′ ( A ) coinciding with S + and E + as functors, respec-tively. We obtain a new tensor structure on the identity functor F A : Rep ′ ( A ) → Rep( A ) solving now the equation for the tensor structures obtained from EF ′ = F A E ′ . This gives aweak quasi-tensor structure to the forgetful functor Rep ′ ( A ) → Hilb, and therefore Rep ′ ( A )becomes unitarily tensorially equivalent to Rep + ( A ′ ) where A ′ is a new unitary weak quasi-bialgebra compatible with the original C ∗ -algebra A , thanks to Theorem 10.5. It followsfrom Prop. 5.2 that A ′ as a weak quasi-bialgebra is only varying by a twist of A . ThereforeRep + ( A ′ ) is unitarily tensor equivalent to Rep + ( A ) by Prop. 9.7 and 9.8, and finally to C + . (cid:3) It follows in particular from the previous characterization that if the tensor structure of C is C*-transportable to C + then C is tensor equivalent to a tensor C ∗ -category, namelyRep + ( A ). We next show more interestingly that the converse implication holds. Thefollowing result will find important applications in the categories arising from affine vertexoperator algebras, Sect. 17. Theorem 12.7.
Let F : C + → C satisfy a) and assume that C admits a weak dimensionfunction (e.g. C is a finite semisimple tensor category). If C is tensor equivalent to a tensor C ∗ -category D + , then the tensor structure of C is C*-transportable to C + in a unique way upto unitary tensor equivalence. Moreover in this way C + becomes unitarily tensor equivalentto D + .Proof. Let D be a weak dimension function on C , and G : D + → C a tensor equivalence.Then D ′ ( ρ ) := D ( G ( ρ )) is a weak dimension function on D + since G ( ρ ⊗ σ ) is isomorphicto G ( ρ ) ⊗ G ( σ ) and D is isomorphism invariant. We may then construct a faithful ∗ -functorof C*-categories D + → Hilb corresponding to D ′ and a weak quasi-tensor structure on thecomposite D + → Hilb → Vec. By Tannaka-Krein duality, see Theorem 10.5, the algebra A of natural transformations of this functor becomes a unitary weak quasi bialgebra,with a corresponding involutive structure ( ∗ , Ω) and such that Rep + ( A ) is unitarily tensorequivalent to D + . Let G ′ : C → D + be an inverse tensor equivalence of G and let E be thecomposed tensor equivalence C → D + → Rep( A ) where the latter functor is obtained fromthe duality theorem in the tensor linear case, see Theorem 5.6 (or equivalently, forgettingthe C*-structure of A ). We may then pick a factorisation of EF through a ∗ -equivalence E + : C + → Rep + ( A ) and the forgetful functor Rep + ( A ) → Rep( A ) by Prop. 10.9. Let † denote the corresponding involution on A . Since all pre-C*-algebra involutions of A aretwisted from one another, we may find a twist t ∈ A , positive with respect to ∗ , such that t † = t − a ∗ t . We may endow A with the twisted involutive structure ( † , Ω t ) by Prop. 8.14and obtain the complete structure and an associated tensor C ∗ -category Rep + t ( A ). Wehave thus shown that ( A, E + , E ) is a compatible triple for F satisfying the necessary andsufficient condition of Theorem 12.6 of C*-transportability. Thus C + becomes a tensor C ∗ -category unitarily tensor equivalent to Rep + t ( A ) and therefore to D + by Prop. 10.1.In the special case that C is a finite semisimple tensor category, it always admits a weakdimension function by Remark 14.4. (cid:3) EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 69
Positive wdf and amenability
The Grothendieck ring Gr( C ) of a rigid semisimple tensor category C is called amenableif it admits a dimension function satisfying a certain analytic property. Such a function,called amenable, is unique and bounds below any other dimension function, see e.g. [100].In this section we extend the framework to weak dimension functions. We show thatthe amenable dimension function is already unique among weak dimension functions andminimizes them. This gives a weaker criterion for amenability. It follows in particularthat if C is a fusion category the lower bound of weak dimension functions is given bythe Frobenius-Perron dimension, and this was our original motivation for the study ofamenability.Let C be a rigid semisimple tensor category and D a weak dimension function on theGrothendieck ring Gr( C ), see Def. 5.8, that will always be assumed positive and symmetricin this section. As already mentioned, we first aim to introduce a notion of amenability for D extending the usual amenability for a genuine dimension. To do this, we closely followthe treatment in Sect. 2.7 in [100], dropping the unitarity assumption on C . Therefore for ρ ∈ Irr( C ) let Λ ρ be the operator of left multiplication by ρ on the complexified algebraGr C ( C ) := Gr( C ) ⊗ Z C . It follows from associativity of Gr( C ) thatΛ ρ Λ σ = X τ m τρ,σ Λ τ , (13.1)with m τρ,σ = dim( τ, ρ ⊗ σ ) and therefore Λ linearly extends to a representation of Gr( C ). Proposition 13.1.
Let C be a rigid semisimple tensor category admitting a weak dimensionfunction. The operator Λ ρ extends to a bounded linear operator on ℓ (Irr( C )) . We have k Λ ρ k ≤ D ( ρ ) for ρ ∈ Irr( C ) and for every weak dimension function D .Proof. The proof extends the corresponding proof for dimension functions, see Prop. 2.7.4in [100], with the modification that u σ = v σ = D ( σ ) is replaced by u σ = D ( σ ) and v σ = (Γ u ) σ = D ( ρσ ) D ( ρ ) ≤ D ( σ ) which implies Γ t ( v ) σ ≤ D ( ρσ ) D ( ρ ) ≤ D ( σ ) = u σ and in turnreplaces Γ t ( v ) = u . Note indeed that these modifications are still compatible with Lemma2.7.3 in [100] and the proof may be completed. (cid:3) Given a dimension function D we consider operators λ µ = P ρ ∈ Irr( C ) µ ( ρ ) D ( ρ ) Λ ρ associated toprobability measures µ on Irr( C ) and then we find that a composition λ µ λ ν = λ µ ∗ ν , with µ ∗ ν the convolution measure defined as at page 71 in [100], µ ∗ ν ( τ ) = X ρ,σ ∈ Irr( C ) m τρ,σ D ( τ ) D ( ρ ) D ( σ ) µ ( ρ ) ν ( σ ) , with m τρ,σ the multiplicity of τ in ρ ⊗ σ . For a weak dimension function a similar formulaholds but µ ∗ ν may not be a probability measure. Indeed k µ ∗ ν k = P τ ∈ Irr( C ) µ ∗ ν ( τ ) = P σ,τ ∈ Irr( C ) µ ( σ ) ν ( τ ) D ( στ ) D ( σ ) D ( τ ) ≤
1. Thus if Irr( C ) is countable and if µ and ν have support Irr( C )then µ ∗ ν is a probability measure precisely when D is a genuine dimension function. Therefore we more generally consider the operators λ µ for any positive measure µ with k µ k ≤
1. One has k λ µ k ≤ k µ k , so k λ µ k = 1 is possible only if µ is a probability measure. Proposition 13.2.
Let D be a weak dimension function on Gr( C ) . Then the followingproperties are equivalent. (a) 1 ∈ Sp λ µ for every probability measure µ , (b) k λ µ k = 1 for every probability measure µ , (c) (ˇ µ ∗ µ ) n ( ι ) /n → for every probability measure µ , with ˇ µ ( ρ ) = µ ( ρ ) , (d) there is a net ξ α ∈ ℓ (Irr( C )) of positive unit vectors such that k Λ ρ ξ α − D ( ρ ) ξ α k → for all ρ ∈ Irr( C ) .If they hold then D is a dimension function.Proof. The equivalence of properties (a)–(d) may be proven just as in the case of ordinarydimension functions, cf. Lemma 2.7.5 in [100], taking into account the slight modificationsmentioned before the statement. The last statement follows from the observation thatΛ is a representation of Gr( C ) in the sense of (13.1), and a 3 ε -argument applied to thevanishing net (Λ ρ (Λ σ − D ( σ ))) ξ α with ξ α as in (d). (cid:3) We recall the definition of amenability.
Definition 13.3.
A dimension function on Gr( C ) satisfying the equivalent properties ofProp. 13.2 is called amenable. The ring Gr( C ) is called amenable if it admits such afunction.The following result extends to weak dimension functions the uniqueness result knownfor an amenable dimension function, see Prop. 2.7.7 in [100]. Theorem 13.4.
An amenable dimension function on
Gr( C ) is unique among weak dimen-sion functions satisfying the equivalent properties of Prop. 13.2 and is given by D ( ρ ) = k Λ ρ k for ρ ∈ Irr( C ) . Any other weak dimension D ′ satisfies D ′ ( ρ ) ≥ D ( ρ ) for all ρ .Proof. The first statement follows from Prop. 13.2 and property (b) applied to the prob-ability measures with support a single irreducible. The second part follows from the firstand Prop. 13.1. (cid:3)
It is well known that amenability can be completely stated in terms of the followingproperty of the left regular representation. The ring Gr( C ) is amenable if and only if forall ρ ∈ Irr( C ), Λ ρ is bounded and k P i µ i Λ ρ i k = P i µ i k Λ ρ i k for finite linear combinationsof basis elements with positive coefficients. These conditions are clearly necessary as bythe previous theorem the amenable dimension function is unique and explicitly given by k Λ ρ k . Conversely, when Λ ρ is bounded, we define the operators λ µ as before with k Λ ρ k inplace of D ( ρ ), ρ ∈ Irr( C ). Then it is easy to see using continuity of µ ∈ ℓ (Irr( C )) → λ µ ∈ B ( ℓ (Irr( C ))) that the positive linearity of k Λ ρ k is equivalent to property (b) of Prop. 13.2.It follows that the linear extension of k Λ ρ k is automatically an amenable weak dimensionfunction by submultiplicativity of the norm. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 71
For example, is also well known that every fusion category C is amenable, and moreoverhas a unique positive dimension function, the Frobenius-Perron dimension determined byFPdim( ρ ) = k Λ ρ k . Indeed, (d) of Prop. 13.2, has a solution given by the vector withcoordinates the dimensions of the simple objects, and by Sect. 8 in [38] or Chapter 4 in[37], k Λ ρ k is indeed a dimension function on Gr( C ). Corollary 13.5. If C is a fusion category then D ( ρ ) ≥ FPdim( ρ ) for every weak dimensionfunction D on Gr( C ) . Another important class of examples is that for which Gr( C ) is commutative. Yamagamishowed that Gr( C ) is amenable if and only if Λ ρ is bounded, see Theorem 3.5 in [135]. Remark 13.6.
The examples that we have studied in the paper show that there may bemore than a natural choice of integral weak dimension functions associated to a fusioncategory. For example, for the pointed fusion categories arising from quantum groups atroots of unity (or vertex operator algebras) at the minimal root (level), FPdim( g ) = 1 onevery irreducible object g , so FPdim is already an integral dimension function. Anothernatural choice is associated to Wenzl functor or, via Finkelberg theorem, to Zhu’s functor.Consider for each level k , Gr( C ( g , q, ℓ ))) for q = e iπ/ℓ , ℓ = d ( k + ˇ h ) and regard it as aquotient of the classical representation ring R ( g ) associated to g . Then the sequence D k of weak dimension functions on Gr( C ( g , q, ℓ )) defined by Wenzl’s functor defines in thepointwise limit the classical dimension function of R ( g ), which is also the unique amenabledimension function of this based ring.We next apply Theorem 13.4 to a weak tensor functor between tensor C ∗ –categoriesstudied in Sect. 3 and we find a useful upper and lower bound for the associated weakdimension function. Corollary 13.7.
Let C and C ′ be rigid tensor C ∗ -categories such that Gr( C ) is amenable.Then every weak tensor ∗ -functor F : C → C ′ defined by F and G satisfies D ( ρ ) ≤ d ′ ( F ( ρ )) ≤ k F ρ,ρ kk G ρ,ρ k d ( ρ ) , ρ ∈ C , where D is the amenable dimension of Gr( C ) , d , d ′ are the intrinsic dimensions of C and C ′ respectively.Proof. Note that the weak dimension function ρ → dim( F ( ρ )) is symmetric as F ( ρ ) is aconjugate of F ( ρ ) by Prop. 3.6. The lower bound then follows from Theorem 13.4. Forthe upper bound see Cor. 3.8. (cid:3) We conclude the paper with a result concerning a dimension preserving property ofunitary weak tensor functor between rigid C ∗ -tensor categories in the amenable case. Thisresult extends a known property for unitary tensor functors, see Cor. 2.7.9 in [100] andreferences therein. Corollary 13.8.
Let C and C ′ be rigid tensor C ∗ -categories with intrinsic dimensions d and d ′ respectively, and let ( F , F, G ) : C → C ′ be a unitary weak tensor ∗ -functor. If theintrinsic dimension d of C is amenable (e.g. C is a fusion category) then d ( ρ ) = d ′ ( F ( ρ )) for all ρ . In particular, when C ′ = Hilb then d ( ρ ) = dim( F ( ρ )) and therefore F is alreadytensor.Proof. By assumption F ∗ and G are isometric, so k F ρ,σ k = k G ρ,σ k = 1. It follows fromCor. 13.7 that d ′ ( F ( ρ )) = d ( ρ ) as d ( ρ ) is the unique amenable dimension function. Inparticular when C ′ = Hilb then ρ → dim( F ( ρ )) is a genuine dimension function and thisimplies that F is tensorial. (cid:3) By the previous corollary when the range category for a weak quasitensor functor is Hilbthen the properties of unitarity and weak tensoriality may coexist only when the functoris automatically tensorial and the intrinsic dimension takes integral values. Thus whena specific ∗ -functor F : C → Hilb on a fusion category is given such that the intrinsicdimension differs from the associated vector space dimension then F admits no unitaryweak tensor structure ( F, G ). On the other hand we know that non-unitary weak tensorstructures exist. For example, this applies to the functor W on C ( s l N , q ) at level k ≥
1. Alternatively, unitary weak quasitensor structures may easily be obtained via polardecomposition.14.
Constructing integral wdf and uniqueness of unitary tensorstructure
In [49] Galindo asks whether a fusion category may admit more than a unitary structuremaking it into a unitary tensor category. In [50] the authors solve the problem in somespecial cases, e.g. pointed and weakly group theoretical categories, and show in these casesa stronger property called complete unitarity. A proof has been given by Reutter in [109]with different methods. The following corollary of Theorem 12.7 gives a positive answerto Galindo’s question for a wide class of tensor categories with possibly infinitely manysimple objects. Note that we do not assume rigidity.
Corollary 14.1.
Let C and C be tensor equivalent C ∗ -tensor categories endowed with anintegral weak dimension function (e.g. they are finite semisimple tensor categories). Then C and C are also unitarily tensor equivalent.Proof. It follows from Theorem 12.7 with C = C + = C , F identity, and C = D + . (cid:3) In Sect. 5 we have remarked about the role of integral weak dimension functions forsemisimple tensor categories in relation to Tannaka-Krein duality and weak quasi-Hopfalgebras. Moreover in Sect. 12 we have used them to turn C ∗ -categories into tensor C ∗ -categories. We next show how to construct these functions for a wide classes of categories. Proposition 14.2.
Let C be a semisimple tensor category and d be a positive (symmetric)weak dimension function on Irr( C ) taking values ≥ . Then for any integer M ≥ , D ( ρ ) = M ⌊ d ( ρ ) ⌋ ρ ∈ Irr( C ) , ρ = ι , defines an integral (symmetric) weak dimensionfunction. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 73
Proof.
We need to show (5.2) for any pair of non-trivial objects ρ , σ ∈ Irr( C ). We have X τ ∈ Irr( C ) D ( τ )dim( τ, ρ ⊗ σ ) ≤ X M ⌊ d ( τ ) ⌋ dim( τ, ρ ⊗ σ ) ≤ X M ⌊ d ( τ )dim( τ, ρ ⊗ σ ) ⌋ ≤ M ⌊ d ( ρ ) d ( σ ) ⌋ ≤ M ( ⌊ d ( ρ ) ⌋ + 1)( ⌊ d ( σ ) ⌋ + 1) ≤ M ⌊ d ( ρ ) ⌋⌊ d ( σ )) ⌋ = 4 M D ( ρ ) D ( σ ) ≤ D ( ρ ) D ( σ ) . (cid:3) Thus all we need to construct integral weak dimension functions is a positive weakdimension function, and we then ask when such a function exists and how to construct it.By Prop. 13.1 a necessary condition is that the operators Λ ρ of left regular representationof Gr C ( C ) on ℓ (Irr( C )) are bounded. This is also a sufficient condition when Gr C ( C ) iscommutative by Theorem 3.5 in [135]. In the general case, we describe two more classesof examples. Theorem 14.3.
Any semisimple rigid C ∗ -tensor category or any semisimple rigid tensorcategory with amenable fusion ring (e.g. any fusion category) admits a natural positivesymmetric dimension function, and therefore infinitely many integral symmetric weak di-mension functions.Proof. The categories in the statement are all known to admit positive symmetric dimen-sion functions, they are respectively given by the intrinsic dimension [82], the norm ofthe left regular representation, see [100] and also Sect 13. Fusion categories are amenableand the Frobenius-Perron dimension is the unique positive dimension of the representationring, cf. Cor. 2.7.8 in [100] and [38]. (cid:3)
Remark 14.4.
The previous result for fusion categories was observed in [87, 116, 59].More precisely, a semisimple tensor category C with finitely many inequivalent simpleobjects always admits positive integral weak dimension functions and when C is a fusioncategory then D may be chosen symmetric. An example is given by the function takingconstant value Max ρ,σ P τ ∈ Irr( C ) dim( τ, ρ ⊗ σ ) for non-trivial ξ ∈ Irr( C ) [116]. Note that anyother integer larger than the constant value of the previous remark defines another weakdimension function and this immediately shows that a fusion category admits infinitelymany weak dimension functions.It follows from Theorem 13.4 that when C is a semisimple rigid tensor category withamenable fusion ring then every symmetric positive integral weak dimension functionbounds from above the amenable dimension. This interesting bound together with theresults of this section shows the great flexibility of weak quasi-Hopf algebras for this classof categories. Examples of fusion categories with different natural integral wdf
Motivated by Remark 14.4, it is natural to ask whether a given fusion category C mayadmit more than one weak integral dimension function corresponding to a w-Hopf algebra.In this subsection we construct examples indicating that this eventuality occurs. Thefirst class of examples is associated to pointed fusion categories over the cyclic group Z N and relies on the basic example A W ( s l N , q, ℓ ) for the minimal value of ℓ . The secondexample shows that already for Z there are infinitely many weak dimension functions ofthis kind, and are obtained using the general constructions of Sect. 6. We shall need theribbon structure naturally associated to the R -matrix of C ( s l N , q, ℓ ). These formulas willbe recalled in the next section. Example 15.1.
Let G be a finite group and ω ∈ H ( G, C × ). The pointed fusion categoryVec ωG admits the natural dimension function taking value 1 on every irreducible and theassociated quasi Hopf algebra is Fun( G, C ) ω , see Example 5.13.6 in [37]. In particular, weobtain a Hopf algebra if and only if ω is trivial in H ( G, C × ). We next see that for G = Z N and ω = 1 for N odd ( ω = − N even) this fusion category may also be described asthe representation category of A = A W ( s l N , q, N + 1). In other words, if g denotes thenatural generator of Z N , D ( g ) = N corresponds to a w-Hopf algebra.Consider the fusion category C ( s l N , q, ℓ ) for q = e iπ/N +1 and let X denote the ob-ject corresponding to the vector representation of U q ( s l N ). We have d ( X ) = 1 and theGrothendieck ring Gr( C ( s l N , q )) is ZZ N with basis given by the objects X = X Λ , . . . , X Λ N − corresponding to the fundamental weights. The fusion rules are given by X k = X Λ k for k ≤ N − X N = 1 [78]. It follows that C ( s l N , q ) is tensor equivalent to Vec ω Z N forsome ω ∈ H ( Z N , T ), cf. Ex. 5.12. Hence in particular Vec ω Z N admits a weak dimensionfunction as required, and we are left to determine ω . The group H ( Z N , T ) is isomorphicto the cyclic group Z N , that we write in multiplicative notation. An explicit isomorphismassociates the N -th root of unity w to the 3-cocycle ω is given by (18.1). For the category C ( s l N , q, ℓ ) the corresponding w may be determined following the procedure at the end ofpage 126 in [78]. In this case, the middle map is identity since the category is strict. Takinginto account the equation appearing in Prop. A.5 in [23] with the additional informationthat S is an isomorphism for the minimum value of the level, we find that ω = 1 for N odd and ω = − N even.Alternatively, we may determine ω in a more direct way as follows. On one hand itis not difficult to see that the only possible values are ω = ±
1. (We shall see a moregeneral statement for higher levels in Prop. 18.9.) On the other, by the the the generalcriterion in Exercise 8.4.11 (iii) pag. 206 in [37], if a pointed fusion category Vec ωG isbraided with braiding c then ω = 1 if and only if for any element γ ∈ G of order somepower of 2, say 2 r , the associated quadratic form q ( γ ) = c ( Y, Y ), with Y simple of class γ , is of order ≤ r . This immediately leads to triviality of ω if N is odd. For N evenwe use the fact that q ( X Λ k ) equals the ribbon structure θ X Λ k , see subsect. 16.1, and that θ X Λ k = q k ( N − k )( N +1) N = e iπk ( N − k ) N , by the proof of Prop. 18.15. Writing N = 2 r h with h anodd integer, it follows that Λ h has order Nh = 2 r but q ( X Λ h ) p = 1 for all 1 ≤ p ≤ r . EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 75
Example 15.2.
We give examples of infinitely many weak dimension functions corre-sponding to w-Hopf algebras on the fusion categories Vec ω Z . They are given by D ( g ) =2 h + 1 for Vec Z and D ( g ) = 2 h for Vec − Z , for h ≥
1, with g the group generator and ω ∈ H ( Z , T ) ≃ Z .Consider the fusion category C ( s l , q, ℓ ) with q = e iπ/ℓ and ℓ ≥
3, and the associatedGrothendieck ring (the Verlinde ring) R ,ℓ with basis given by the equivalence classes ofselfconjugate irreducible objects X = I , X , . . . , X k . Fusion rules are given by X i X j = P min { i,j } max { i + j − k, } X i + j − r , see [22, 37]. The element X = X k satisfies X = I , so it generatesa pointed full fusion subcategory C k ≃ Vec ω Z . We determine ω ∈ {± } by means ofEx. 8.4.11 iii) in [37] again, so in this case ω = 1 precisely when the quadratic form q ( g ) = c ( X, X ) associated to restricted braiding of C k satisfies q ( g ) = 1 or q ( g ) = 1.Arguments similar to those of the previous example give q ( g ) = θ X , with θ the usualribbon structure of c , whose value on X = X k is θ X = q k ( k +2) / = e iπk/ cf. Prop. 18.15.It follows that ω = 1 if and only if k is even. On the other hand, C ( s l , q, ℓ ) is tensorequivalent to the representation category of A W ( s l , q, ℓ ) so C k is tensor equivalent to aquotient w-Hopf algebra A → B k by Cor. 6.11. Since X corresponds to a representationof A of dimension k + 1, we have B k = C ⊕ M k +1 ( C ) and a weak dimension function D on C k , and therefore on Vec ω Z as required. Example 15.3.
The methods of the above examples may be combined to construct moreexamples of w-Hopf algebras. a) For example, if g ∈ Z N is the natural generator, for k ≤ N − g k generates a cyclic subgroup of order M = N gcd { k,N } . Therefore the fullsubcategory of C ( s l N , q, ℓ ) for q = e iπ/N +1 generated by X Λ k , which is pointed over Z M ,corresponds to a quotient of A W ( s l N , q, N + 1) (with dimension of the natural generatorof Z M given by D ( h ) = (cid:0) Nk (cid:1) ) and also to A W ( s l M , q, M + 1) (with dimension D ′ ( h ) = M )with a possibly twisted associator. b) The even subcategory of C ( s l , q, ℓ ) for q = e iπ/ℓ isan example of non-pointed full fusion subcategory, and therefore it gives rise to a quotientw-Hopf algebra B = C ⊕ M ⊕ M . . . . c) More information on full fusion subcategories of C ( s l N , q, ℓ ) for q = e iπ/ℓ may be found in [117]. Remark 15.4.
Ribbon structures first appeared as statistics phases for WZW and cosetmodels in conformal field theory. Some formulae for the statistics phases, including theautomorphism case of interest in Ex. 15.2, have been generalized by Rehren in the frame-work of conformal nets. Most importantly, in that paper the author derives the axiomsof a modular category extending previous work for certain conformal models [106] andreferences therein. The ribbon structure in the conformal net approach to CFT is givenby θ X = e πih X with h X the minimal eigenvalue of the conformal Hamiltonian L in theirreducible representation X , by the conformal spin and statistics theorem [58]. In theframework of vertex operator algebras one has an analogous formula [61, 62, 63]. Remark 15.5.
In the setting of rigid tensor C ∗ -categories with infinitely many simpleobjects, Van Daele and Wang constructed compact quantum groups A o ( F ) associated toan invertible matrix F with rk( F ) ≥ q (2) for rk( F ) = 2. For a given q > Rep( A o ( F )) turns out to be tensor equivalent to Rep(SU q (2)) with q suitably determinedby F [6, 7]. It follows that Rep(SU q (2)) admits the (non-weak) dimension function takingthe generating representation to the rank of F . Note that only finitely many (non-weak)integral dimension functions arise in this way. This follows from the fact that rk( F ) isbounded above by the quantum dimension [131]. In this setting, it is important to recallthe remarkable work by Neshveyev and Yamashita on the classification of compact quantumgroups that beyond the fusion rules, share the integral dimensions with a given compactsimple simply connected Lie group G , see [101] and references therein.16. Quantum groups at roots of unity, fusion categories and unitaryribbon wqh algebras via wdm
Let g be a complex simple Lie algebra and q a primitive complex root of unity. We denoteby ℓ the order of q . Let U q ( g ) be the quantized universal enveloping algebra in the senseof Lusztig, see below for a definition and references. It is known that the category of finitedimensional representations of U q ( g ) is not semisimple, but it gives rise to a semisimpleribbon fusion category that we denote by C ( g , q, ℓ ) following [111]. Moreover, the categories C ( g , q, ℓ ) are known to be modular for certain values of q see [2, 3, 51, 108, 111, 114] andreferences therein, see also Subsects. 15 . . C + ( g , q, ℓ ) equivalent to C ( g , q, ℓ ) for certain primitive roots of unity, that we call the minimal roots and precisely define later [79, 128, 134].In this section we construct semisimple weak quasi-Hopf algebras associated to C ( g , q, ℓ )and unitary weak quasi-Hopf algebras associated to C + ( g , q, ℓ ) when q is a minimal root.Our approach may broadly be summarized as follows.From the categories C ( g , q, ℓ ), we construct weak quasi-tensor functors to the categoryof vector spaces and then we use Tannaka-Krein reconstruction to obtain our examples.We shall do this following two alternative approaches, and both turn out useful for usin the study of unitary tensor categories. The first approach goes back to [87, 117, 59]. Itconsists in identifying a certain integral valued weak dimension function D on C ( g , q, ℓ ),and then we apply the abstract reconstruction result, Theorem 5.9. This leads to theconstruction of a ribbon weak quasi-Hopf algebra A ( g , q, ℓ ) corresponding to C ( g , q, ℓ ) whichis defined up to twist and isomorphism. Moreover, when q is a minimal root, we applyTheorem 10.5 and we obtain a unitary structure A + ( g , q, ℓ ) on A ( g , q, ℓ ). By the results ofSect. 12 this general approach addresses the study of unitary structures via the associator.It follows that this viewpoint will turn out fruitful for the construction of unitary ribbonstructures for representation categories of affine VOA in Sect. 17. It perhaps conveys theidea of the amount of information needed to obtain these unitary structures from othersources for which they are known to exist.A second approach consists in identifying a natural functor W : C ( g , q, ℓ ) → Vec asso-ciated to the same dimension function D as before, and thus it is a particular case of theformer, and will be studied in Sect. 24. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 77
When q is a minimal root, the work of [128] shows that U q ( g ) is a Hermitian coboundaryHopf algebra with compatible involution (in a topological sense). We shall introduce thisnotion in Sect. 24 and summarize this result in Theorem 24.1. However, in this section weshall not need to go into these details. We briefly recall the basic results on quantumgroups at roots of unity that we shall need. For a complete presentation we refer to [22, 114]and references therein.Let g be a f.d. complex simple Lie algebra, and q a complex root of unity whose orderwe denote by ℓ ′ . (Thus the order ℓ of q is given by ℓ = ℓ ′ if ℓ ′ is odd and ℓ = ℓ ′ / ℓ ′ is even.) Note that our ℓ has the same meaning in [111], but the roles of ℓ and ℓ ′ areexchanged in [114, 122]. Definition 16.1.
We shall say that the order ℓ pf q is large enough if ℓ > ˆ h when ℓ isnot divisible by d and ℓ > dh ∨ otherwise, with h ∨ the dual Coxeter number of g .Throughout this paper we assume that ℓ os large enough. Let h a Cartan subalgebra, α , . . . , α r a set of simple roots, and A = ( a ij ) the associated Cartan matrix. Consider theunique invariant symmetric and bilinear form on h ∗ such that h α, α i = 2 for a short root α and let θ denote the highest root. Let E be the real vector space generated by the rootsendowed with its euclidean structure h x, y i . Let Λ be the weight lattice of E and Λ + thecone of dominant weights.Consider the complex ∗ –algebra A = C [ x, x − ] of Laurent polynomials with involution x ∗ = x − , and let C ( x ) be the associated quotient field, endowed with the involutionnaturally induced from C [ x, x − ]. We consider Drinfeld-Jimbo quantum group U x ( g ), i.e.the algebra over C ( x ) defined by generators E i , F i , K i , K − i , i = 1 , . . . , r , and relations K i K j = K j K i , K i K − i = K − i K i = 1 ,K i E j K − i = x h α i ,α j i E j , K i F j K − i = x −h α i ,α j i F j ,E i F j − F j E i = δ ij K i − K − i x d i − x − d i , − a ij X ( − k E (1 − a ij − k ) i E j E ( k ) i = 0 , − a ij X ( − k F (1 − a ij − k ) i F j F ( k ) i = 0 , i = j, where d i = h α i , α i i /
2, and, for k ≥ E ( k ) i = E ki / [ k ] d i !, F ( k ) i = F ki / [ k ] d i !. Note that d i is an integer, hence so is every inner product h α i , α j i . Quantum integers and factorialsare defined in the usual way, [ k ] x = x k − x − k x − x − ; [ k ] x ! = [ k ] x . . . [2] x , [ k ] d i = [ k ] x di , and resultselfadjoint scalars of C ( x ). There is a unique ∗ –involution on U x ( g ) making it into a ∗ –algebra over C ( x ) such that K ∗ i = K − i , E ∗ i = F i . This algebra becomes a Hopf algebra, with coproduct, counit, and antipode defined, asfollows, see e in [128], where his ˜ K i corresponds to our K i , see also [22, 114],∆( K i ) = K i ⊗ K i , ∆( E i ) = E i ⊗ K i + 1 ⊗ E i , ∆) F i ) = F i ⊗ K − i ⊗ F i ,S ( K i ) = K − i , S ( E i ) = − E i K − i , S ( F i ) = − K i F i ,ε ( K i ) = 1 , ε ( E i ) = ε ( F i ) = 0 . One has the following relations between coproduct, antipode and involution for a ∈ U x ( g ), ∆( a ∗ ) = ∆ op ( a ) ∗ (16.1) ε ( a ∗ ) = ε ( a ) , S ( a ∗ ) = S ( a ) ∗ , S ( a ) = K − ρ aK ρ , (16.2)where 2 ρ the sum of the positive roots, and, for an element α = P i k i α i of the root lattice, K α := K k . . . K k r r .Following [114], order to construct an R -matrix, we need to embed the original algebrainto a larger algebra, and we first need to extend the ring of scalars from A to A ′ := C [ x /L , x − /L ] , with L the smallest positive integer such that L h λ, µ i ∈ Z for all dominant weights λ , µ .The values of L for all Lie types are listed in [114]. For example, L = N for g = s l N .We define the integral form U A ′ ( g ) as the A ′ –subalgebra generated by the elements E ( k ) i , F ( k ) i and K i . This is known to be a ∗ –invariant Hopf A ′ –algebra with the structure inheritedfrom U x ( g ). Applying the construction in Sect. 1 of [114] to the modified polynomial ring,we may replace U A ′ ( g ) with an extended topological ribbon Hopf algebra U † A ′ ( g ).We fix q ∈ T a primitive root of unity of order ℓ ′ and we set ℓ ′ = ∞ if q n = 1 for all n ∈ N . We consider the ∗ –homomorphism A ′ → C which evaluates x /L to a specifiedcomplex L -root q /L of q . We form the tensor product ∗ –algebra, U q ( g ) := U † A ′ ( g ) ⊗ A ′ C . The algebra U q ( g ) becomes a ribbon complex Hopf algebra with a ∗ –involution, and istopological in the sense of [114]. Note that the R -matrix R and the ribbon element v ∈ U q ( g ) depend only on the choice of q /L , see Sect. 1 in [114], Sect. 1.4 in [128]. Furthermore,a square root w ∈ U q ( g ) of v is well defined up to a sign choice in every representationentering the definition of U q ( g ), we refer to Sect. 1 in [114] for details. C ( g , q, ℓ ) . In this subsection we assume ℓ ′ < ∞ . Constructionsdue to [2, 3, 51, 108], give rise to a semisimple, ribbon, fusion category, C ( g , q, ℓ ) that webriefly outline. Notice that the constructions impose no restriction on the order of q , anddepend on the order of q .Let d denote the ratio between the squared lengths of the longest to the shortest root, so d = 1 for Lie types ADE , d = 2 for BCF and d = 3 for G . An irreducible highest weightmodule of dominant weight λ of the classical algebra admits a natural deformation to amodule of U q ( g ), denoted V λ and called Weyl module, which may fail to be irreducible Sect.1, 3 [114], Ch. 11.2 [22]. Andersen developed the notion of tilting module. Informally,direct sums are replaced by Weyl filtrations, and in this way the category T ( g , q, ℓ ) oftilting modules becomes a tensor category [22, 114]. More precisely, by Cor. 5 in [114]every tilting module decomposes into a direct sum of indecomposable tilting modules, EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 79 and every indecomposable tilting module is isomorphic to a unique indecomposable tiltingmodule T λ with maximal vector of weight λ , with λ ∈ Λ + . Thus T λ has a filtration bysubmodules 0 ⊂ V λ ⊂ V ⊂ V ⊂ · · · ⊂ T λ such that V /V λ ≃ V µ , V /V ≃ V ν , . . . with λ > µ > ν . . . , [22] p. 363, and the dual T ∗ λ has a similar filtration and is isomorphicto T − w λ . It follows from the linkage principle that every Weyl module V λ is tilting andcoincides with T λ when λ lies in the closed Weyl alcove Λ ℓ := { λ ∈ Λ + : h λ, θ + ρ i ≤ ℓ } bye.g. [4] Subsect. 1.1 and irreducible by Prop. 2.4 in [128].It follows from Sebsect. 15.1, see also Theorems 3, 4 in [114], that the category of tiltingmodules over U q ( g ) is a ribbon category. For a fixed choice of q /L , the corresponding R -matrices define corresponding braided symmetries for the representation category, formore details on the classification in the type A case, and references see Sect. 18. Theribbon structure v of the category of tilting modules is given by v λ = q h λ,λ +2 ρ i for λ ∈ Λ ℓ ,with Λ ℓ := { λ ∈ Λ + : h λ, θ + ρ i < ℓ } the open Weyl alcove.For a detailed description of the following quotient construction, we refer to Gelfand andKazhdan [51]. Every object of T ( g , q, ℓ ) decomposes into a direct sum of indecomposablesubmodules, and this decomposition is unique up to isomorphism. One can form two fulllinear subcategories, T , and T ⊥ of T ( g , q, ℓ ), with objects, respectively, those representa-tions which can be written as direct sums of V λ , with λ ∈ Λ ℓ only, and those which haveno such V λ as a direct summand.The objects of T ⊥ and T are called negligible and non-negligible, respectively. A mor-phism T : W → W ′ of T ( g , q, ℓ ) is called negligible if it is a sum of morphisms that factorthrough W → N → W ′ with N negligible.The category T ⊥ of negligible modules satisfies the following properties, [2, 51].(1) Any object W ∈ T ( g , q, ℓ ) is isomorphic to a direct sum W ≃ W ⊕ N with W ∈ T and N ∈ T ⊥ .(2) For any pair of morphisms T : W → N , S : N → W of T ( g , q, ℓ ), with N ∈ T ⊥ , W i ∈ T , then ST = 0 . (3) For any pair of objects W ∈ T ℓ ( g ), N ∈ T ⊥ , then W ⊗ N and N ⊗ W ∈ T ⊥ .Property (1) follows easily from the mentioned decomposition of objects of T ( g , q, ℓ ),while property (2) means that no non-negligible module can be a summand of a negligibleone (however, it can be a factor of a Weyl filtration of a negligible).For completeness we recall that negligible indecomposable tilting modules are charac-terized by the property of having zero quantum dimension. A morphism T : W → W ′ isnegligible if and only if Tr W ( ST ) = 0 for all morphisms S : W ′ → W .The category C ( g , q, ℓ ) is defined as the quotient of the category of tilting modulesby the ideal of negligible modules, that is the smallest full subcategory containing theindecomposable tilting modules T λ with λ / ∈ Λ ℓ , with Λ ℓ . More in detail, let Neg( W, W ′ )be the subspace of negligible morphisms of ( W, W ′ ). Then the quotient category, C ( g , q, ℓ ),is the category with the same objects as T ( g , q, ℓ ) and morphisms between the objects W and W ′ the quotient space,( W, W ′ ) C ( g ,q,ℓ ) := ( W, W ′ ) / Neg(
W, W ′ ) . Gelfand and Kazhdan endow C ( g , q, ℓ ) with the unique structure of a tensor category suchthat the quotient map T ( g , q, ℓ ) → C ( g , q, ℓ ) is a tensor functor. The tensor product ofobjects and morphisms of C ( g , q, ℓ ) is usually denoted by W ⊗ W ′ and S ⊗ T respectively,and referred to as the truncated tensor product in the physics literature. This is now asemisimple tensor category and { V λ , λ ∈ Λ ℓ } is a complete set of irreducible objects.The ribbon structure of C ( g , q, ℓ ) is induced by that of the tilting category. Also theformulas for the fusion coefficients and quantum dimensions are well-known, and regulatedby the affine Weyl group, as in Sect. 2, 5 of [114], but we shall only need them in somespecial cases later on, so we refrain from recalling them in full generality. However, it willbe important for us to recall that C ( g , q, ℓ ) depends on q but the Grothendieck semiring R ( C ( g , q, ℓ )) depends only on ℓ . We shall refer to R ( C ( g , q, ℓ )) as the Verlinde fusion ring. Further properties of modularity C ( g , q, ℓ ) depend onon the choice of q /L as a primitive root of unity of order ℓ ′ L and on the order ℓ ′ of q . Werefer to the papers by Rowell and Sawin [111, 114] for a detailed treatment. For examplethe cases where 2 d | ℓ ′ give modular categories and this is the case of most physical interest,and also that meeting the purpose of our paper.More in particular, we shall mostly be interested in the“minimal roots” q = e iπ/ℓ , q /L = e iπ/ℓL , d | ℓ. Indeed in this case C ( g , q, ℓ ) is equivalent to a unitary ribbon fusion category that wedenote by C + ( g , q, ℓ ) by [128, 134], and indeed modular. A ( g , q, ℓ ) . We introduce the function D on the Grothendieck ring of C ( g , q, ℓ ), which assigns the vector space dimension of thecorresponding representation of g to each irreducible λ ∈ Λ ℓ . It follows easily from thequotient construction and from the fact that every tilting module decomposes uniquelyup to isomorphism into a direct sum of indecomposable tilting modules, that D is indeeda weak dimension function on C ( g , q, ℓ ). We shall refer to it as the classical dimensionfunction . We may then apply Theorem 5.9 and we have, up to isomorphism and twist, afinite dimensional weak quasi Hopf C ∗ -algebra A ( g , q, ℓ ). We next fix a root of unity ofthe form q = e iπ/ℓ with d | ℓ . Then by [128, 133], and Theorem 10.10 A ( g , q, ℓ ) becomes aunitary weak quasi-Hopf algebra.17. VOAs, the Zhu algebra and conformal nets
In this section we briefly describe some natural constructions of weak quasi-Hopf algebrasand w-Hopf algebras from the theory of vertex operator algebras (VOAs) and the theory ofconformal nets. This leads to some interesting questions and possible future applicationsof the theory described in this article. We will restrict to VOAs and conformal nets whoserepresentation category are known to be modular tensor categories. These are the rationalVOAs satisfying the assumptions in [63] and the completely rational conformal nets firstdefined and studied in [75].
EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 81
We start with the VOA case. Let V be a VOA satisfying the rationality assumptions in[63], namely:(a) V is simple and of CFT type, the contragredient module V ′ is isomorphic to V asa V -module;(b) every Z ≥ -graded weak V -module is a direct sum of irreducible V -modules;(c) V is C -cofinite.Then, by the results in [63], the category Rep( V ) of V -modules has a natural structureof modular tensor category.We wish to define a canonical functor F V : Rep( V ) → Vec. When the assumption inTheorem 5.9 are satisfied then, thanks to the Tannaka-Krein duality result in Theorem 5.6we will be able to associate a weak quasi-Hopf algebra to Rep( V ).Let M be a V -module. In the following we will denote by Y M ( a, z ) = X n ∈ Z a M ( n ) z − n − , a ∈ V (17.1)the vertex operators on M . If ν ∈ V is the conformal vector we write Y M ( a, z ) = X n ∈ Z L Mn z − n − , a ∈ V . (17.2)In particular L M denotes the conformal Hamiltonian on M .The rationality assumptions for V imply that M can be written as a finite direct sum M = M i M i (17.3)of irreducible V-modules M i . For each M i there is a (necessarily unique) complex number h i such that M i = M n ∈ Z ≥ Ker( L M i − ( h i + n )1 M i ) (17.4)with M i ( n ) = Ker( L M i − ( h i + n )1 M i ) and M i (0) = { } . Note that every M i ( n ) is finitedimensional. We now define a finite dimensional subspace M (0) ⊂ M by M (0) := M i M i (0) . (17.5)It is easy to see that M (0) is independent from the choice of the direct sum decompositionin Eq. (17.3). Moreover, it can be shown that U ( M, V ) M (0) = M where U ( M, V ) is thesubalgebra of End( M ) generated by the vertex operator coefficients a M ( n ) , a ∈ V , n ∈ Z .Now let M α and M β be V -modules M α and M β and T : M α → M β a V -modulehomomorphism. From the equality T L M α = L M β T it follows that T M α (0) ⊂ M β (0) .We now define a linear functor F V : Rep( V ) → Vec in the following way. If M is anobject in Rep( V ), i.e. a V -module, then F V ( M ) = M (0) . If T : M α → M β is a morphismin Rep( V ), i.e. a V -module homomorphism, then F V ( T ) = T ↾ M α (0) . If F V ( T ) = 0 then, T M α = T U ( M α , V ) M α (0) = U ( M β , V ) T M α (0) = { } so that T = 0 and hence F V is faithful. We are now in the position to apply Theorem 5.6. Let A ( V ) := Nat ( F V ). Note that A ( V )is a semisimple associative algebra and that it can be identified with the Zhu’s algebra of V , [48, 137]. Moreover, there is an equivalence E V : Rep( V ) → Rep( A ( V )) which, aftercomposition with the forgetful functor : Rep( A ( V )) → Vec is isomorphic to F V . Theorem 17.1.
Let V be a VOA satisying the assumptions (a), (b), (c) at the beginningof this section. Assume moreover that M D ( M ) := dim( F V ( M )) , M irreducible, gives aweak dimension function on the modular tensor category Rep( V ) . Then, the Zhu’s algebra A ( V ) admits a structure of a weak quasi-Hopf algebra with a tensor equivalence E V :Rep( V ) → Rep( A ( V )) which, after composition with the forgetful functor : Rep( A ( V )) → Vec is tensor isomorphic to F V .Proof. By Theorem 5.9 F V admits a weak quasi-tensor structure and the conclusion followsfrom Theorem 5.6. (cid:3) Remark 17.2.
The functor E V : Rep( V ) → Rep( A ( V )) already appeared in the literaturewithout mention to the tensor structure, see [28, 66, 137]. Remark 17.3.
The condition on M D ( M ), which we will call the weak dimensioncondition, is not satisfied in general. For example if V is a rational unitary Virasoro VOAthen D ( M ) = 1 for all irreducible V-modules M . Moreover, from the known fusion rulesof these models, see e.g. [76, Sec. 2.2], it follows that one can always find an irreducible M with D ( M ⊗ M ) = 2 > D ( M ) and hence the weak dimension condition is not satisfied.On the other hand the class of rational VOAs satisfying the weak dimension conditioninclude many remarkable examples such as the unitary simple affine VOAs and the latticeVOAs.We now discuss the case of unitary affine VOAs. Let g be a complex simple Lie algebraand let k be a positive integer. Moreover let g R ⊂ g be a real form of g and let G bethe corresponding simply connected compact simple Lie group. We denote by V g k thelevel k affine simple unitary VOA associated to the pair ( g , k ). It is known to satisfy theassumptions (a), (b), (c) so that Rep( V g k ) is a modular tensor category. Accordingly wecan consider the functor F V g k which satisfies the weak-dimension condition so that theZhu’s algebra A ( V g k ) admits a weak quasi-Hopf algebra structure.Now, let us consider the quantum group U q ( g ) with q = e iπd ( k + h ∨ ) , where h ∨ is the dualCoxeter number of g . We denote by ˜ O q ( g ) the semisimplified category obtained from thecategory of tilting modules. It is a modular braided category admitting a compatible C*-structure by [128, 134]. Let F ( g ,q ) : ˜ O q ( g ) → Vec be the Wenzl functor. Then F ( g ,q ) satisfiesthe weak-dimension condition and hence it defines a weak quasi-Hopf algebra b C ( G, k + h ∨ ).By a result of Finkelberg [39, 40], see also [65, Sec. 3], the category ˜ O q ( g ) is tensorequivalent to Rep( V g k ). Since the weak dimension functions for the functors F V g k and F ( g ,q ) have the same range in Z ≥ we can conclude that A ( V g k ) and b C ( G, k + h ∨ ) are, upto a twist, isomorphic weak quasi-Hopf algebras, cf. the discussion after Theorem 5.9. Inparticular, A ( V sl N k ) admits a structure of w-Hopf algebra.
EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 83
We now discuss the unitary aspects of the above constructions. We first need to recallsome properties of the Zhu’s algebra and fix some notation.A homogeneous element a ∈ V of conformal weight d ∈ Z , i.e. such that L a = da andevery V -module M then a Mn is defined by a Mn := a M ( n + d − , n ∈ Z . For a general a ∈ Va Mn is defined by linearity. As a vector space the Zhu’s algebra is a quotient V /O ( V ) fora certain subspace O ( V ) ⊂ V and we denote by a [ a ] the quotient map : V → A ( V ).When V satisfies the assumption (a), (b), (c) then O ( V ) = { a ∈ V : a M ↾ M (0) = 0 for all V -modules M } . (17.6)Moreover, for every V -module M the map [ a ] a M ↾ M (0) is a representation of theassociative algebra A ( V ) on M (0) which is the one corresponding to E V ( M ) in Theorem17.1.Let V be a unitary VOA [20, 29] satisfying the rationality assumptions (a), (b), (c).Note that if V is simple and unitary then a is necessarily of CFT type and isomorphic tothe contragredient module V ′ as a V-module so that (a) is a priori satisfied. Let θ be thePCT operator giving the unitary structure on V . By [41, Eq. 5.3.1] and [28, Prop. 2.3.]the map [ a ] [ e L ( − L a ]is an involutive anti-automorphism of A ( V ). On the other hand, being θ an anti-linearinvolutive automorphism of V , we have that θ ( O ( V )) = O ( V ) and the map [ a ] [ θa ] isan anti-linear involutive automorphism of the associative algebra A ( V ). It follows that[ a ] [ a ] ∗ := [ e L ( − L θa ]is an anti-linear involutive automorphism of A ( V ) i.e. it gives a *-algebra structure on A ( V ) canonically associated to the unitary structure of A ( V ). Proposition 17.4.
Let M be a unitary V -module then the restriction to M (0) of the in-variant scalar product of M makes E V ( M ) into a *-representation of A ( V ) . Moreover, theabove restriction gives a one-to-one correspondence between the invariant scalar producton M and the scalar products making E V ( M ) into a *-representation of A ( V ) .Proof. The first claim follows in a straightforward way from the definition of invariantscalar product and the *-operation on A ( V ). Now, let U ( M, V ) be the associative algebragenerated by the vertex operator coefficients a M ( n ) , a ∈ V , n ∈ Z as before. U ( M, V ) carriesa Z -grading U ( M, V ) = M n ∈ Z U ( M, V ) n where U ( M, V ) n := { X ∈ U ( M, V ) : e itL M Xe − itL M = e itn X } . Accordingly, we have a Mn ∈ U ( M, V ) n . Moreover, for every X ∈ M there is an X ∗ ∈ U ( M, V ) such that ( m , Xm ) = ( X ∗ m , m ) for all m , m ∈ M (0) , where ( · , · ) is theinvariant scalar product on M . Note that ( a Mn ) ∗ = ( e L ( − L θa ) M − n for all a ∈ V and all n ∈ Z so that ( U ( M, V ) n ) ∗ = U ( M, V ) − n for all n ∈ Z . In particular U ( M, V ) is a *-subalgebra of U ( M, V ). For every X ∈ U ( M, V ) we have XM ⊂ M and hence X restrictsto an endomorphism ˜ X of M . Now, given m , m ∈ M we have ( X k m , Y n m ) = 0 if k = n . Accordingly we have( Xm , Y m ) = ( m , X n ∈ Z ( X n ) ∗ Y n m )which shows that the invariant scalar product on M is determined by its restriction to M (0) Now, let ( · , · ) be a fixed invariant scalar product on M and let {· , ·} any scalar producton M (0) making E V ( M ) into a *-representation of A ( V ). Then there is an A ( V )-moduleisomorphism T : M (0) → M (0) such that { m , m } = ( m , T m ) for all m , m ∈ M (0) .Since E V is an equivalence of categories there is a unique V -module map T : M → M such that E V ( T ) = T and we can define a sesquilinear form {· , ·} M on M by { m , m } M =( m , T m ), m , m ∈ M . It is now straightforward to check that {· , ·} M is an invariantscalar product on M whose restriction to M (0) is {· , ·} . (cid:3) Remark 17.5.
Let V a unitary vertex operator algebra satisfying the assumptions (a),(b), (c) so that Rep( V ) is a modular tensor category. Let Rep + ( V ) be the C*-category ofunitary representations of V . Then the forgetful functor : Rep + ( V ) → Rep( V ) is linearequivalence if and only if every V -module is unitarizable. In this case Rep + ( V ) is equiv-alent as a C*-category to the representation category Rep + ( A ( V )) of finite dimensional*-representations of the C*-algebra A ( V ). It is not clear in general if the linear equiva-lence Rep + ( V ) ≃ Rep( V ) can be used to make Rep + ( V ) into a tensor C ∗ -category tensorequivalent to Rep( V ). This is an important problem which has been recently solved insome special cases by B. Gui [56, 57]. We also recall a work by Kirillov on the constructionof a tensor ∗ -category closely related to Rep( V ) which preceded the work by Huang andLepowsky [80]. Proposition 17.6.
Let V a unitary vertex operator algebra satisfying the assumptions (a),(b), (c). Then the equivalence E V : Rep( V ) → Rep( A ( V )) gives in a canonical way a faith-ful *-functor E + V : Rep + ( V ) → Rep + ( A ( V )) . If the forgetful functor Rep + ( V ) → Rep( V ) is an equivalence of linear categories then A ( V ) is a C*-algebra and E + V : Rep + ( V ) → Rep + ( A ( V )) is an equivalence of C*-categories. Moreover, in the latter case, any equiv-alence of linear categories S V : Rep( A ( V )) → Rep( V ) together with an isomorphism η : E V ◦ S V → Rep( A ( V )) gives a canonical *-equivalence S + V : Rep + ( A ( V )) → Rep + ( V ) with E + V ◦ S + V unitarily equivalent to the identity.Proof. Let M be a unitary V -module. Then E + V ( M ) is defined to be the A ( V )-module E V ( M ) together with the scalar product obtained by restricting the given invariant scalarproduct on M . Then, thanks to Prop. 17.4 E + V is a faithful *-functor which becomean equivalence if the forgetful functor : Rep + ( V ) → Rep( V ) is a linear equivalence.. Inthe latter case we have the linear eqivalence Rep( A ( V )) ≃ Rep + ( A ( V )) and hence A ( V )is a C*-algebra. Assume now the linear equivalence Rep + ( V ) ≃ Rep( V ) and let S V :Rep( A ( V )) → Rep( V ) be an equivalence with a natural isomorphism η : E V ◦ S V → Rep( A ( V )) . Let W be a C*-module for A ( V ) and let ( · , · ) W be the corresponding scalar EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 85 product. Then ( η W · η W · ) W is a scalar product on E V ◦ S V ( W ) making it into a *-representation of A ( V ). Then it follows from Prop. 17.4 and the assumption Rep + ( V ) ≃ Rep( V ) that there is a unique invariant scalar product on S V ( W ) which restricts to ( η W · η W · ) W . This scalar product defines a unitary V -module S + V ( W ) and it is not hard to seethat the map W S + V ( W ) defines a functor with the desired properties. (cid:3) Theorem 17.7.
Let V be a unitary vertex operator algebra satisfying assumptions (a),(b), (c) and such that the forgetful functor : Rep + ( V ) → Rep( V ) is a linear equivalenceand assume that the functor F V : Rep( V ) → Vec satisfies the weak dimension conditionin Remark 17.3. Then
Rep + ( V ) admits a structure of tensor C ∗ -category with unitarybraided symmetry such that the forgetful functor : Rep + ( V ) → Rep( V ) is a braided tensorequivalence if and only if the weak quasi-Hopf algebra on A ( V ) obtained from a weak quasi-tensor structure on the functor F V : Rep( V ) → Vec admits the structure of a Ω -involutiveweak quasi-Hopf C*-algebra compatible with the canonical *-structure on A ( V ) .Proof. The functor F + V : Rep + ( V ) → Hilb obtained by composition of the equivalence E + V : Rep + ( V ) → Rep + ( A ( V )) with the forgetful functor : Rep + ( A ( V )) → Hilb is a*-functor as a consequence of Prop. 17.6. If Rep + ( V ) admits a structure of tensor C ∗ -category such that the forgetful functor : Rep + ( V ) → Rep( V ) is a tensor equivalence then F + V admits a weak quasi-tensor *-structure so that Nat ( F + V ) admits the structure of a weakquasi-Hopf C*-algebra as a consequence of Theorem 10.5. By construction the C*-algebra A ( V ) with its canonical *-operation is isomorphic to Nat ( F + V ) so that it inherits from thelatter the structure of a Ω-involutive weak quasi-Hopf C*-algebra coinciding , up to a twist,with the weak quasi-Hopf algebra structure on A ( V ) obtained from a weak quasi-tensorstructure on the functor F V : Rep( V ) → Vec.Conversely, if A ( V ) admits the structure of a Ω-involutive weak quasi-Hopf C*-algebrawith the canonical *-structure then, by Corollary 9.5 Rep + ( A ( V )) is a tensor C ∗ -categorytensor equivalent Rep( A ( V )) and hence to Rep( V ) . Let S V : Rep( A ( V )) → Rep( V ) be anytensor equivalence together with an isomorphism of tensor functors η : E V ◦ S V → Rep( A ( V )) and let S + V : Rep + ( A ( V )) → Rep + ( V ) be the corresponding canonical *-equivalence as inProp. 17.6 so that E + V ◦ S + V unitarily equivalent to the identity.Given unitary V -modules M α , M β ∈ Rep + ( V ) we define a unitary module M α ⊗ M β by M α ⊗ M β := S + V (cid:0) E + V ( M α ) ⊗ E + V ( M β ) (cid:1) . Moreover, if α denotes the unitarty associator in Rep + ( A ( V )) we define the unitaries α ′ M α ,M β ,M γ : ( M α ⊗ M β ) ⊗ M γ → M α ⊗ ( M β ⊗ M γ )by α ′ M α ,M β ,M γ := S + V (1 E + V ( M α ) ⊗ η − E + V ( M β ) ⊗ E + V ( M γ ) ◦ α E + V ( M α ) , E + V ( M β ) , E + V ( M γ ) ◦ η E + V ( M α ) ⊗ E + V ( M β ) ⊗ E + V ( M γ ) )where η : E V ◦ S V → Rep( A ( V )) is the isomorphism used to define the functor S + V . Then, thanks to Prop. 17.6, one can check that this gives the desired C*- tensor structureon Rep + ( V ). From the tensor equivalence Rep + ( V ) ≃ Rep( V ) we see that Rep + ( V ) admitsa braiding making the equivalence a braided tensor equivalence and this braided symmetryon Rep + ( V ) is necessarily unitary by [49]. (cid:3) We can apply the previous corollary directly to the affine type A VOAs. On the otherhand, theanks to Finkelberg’s equivalence, Theorem 17.7 can be used directly to give thesame result for all Lie types.
Theorem 17.8. . Let g be a complex simple Lie algebra and let k be a positive integer andlet V g k be the corresponding level k affine unitary vertex operator algebra. Then Rep + ( V g k ) admit the structure of tensor C ∗ -category with unitary braiding symmetry such that theforgetful functor : Rep + ( V g k ) → Rep( V g k ) is a braided tensor equivalence.Proof. It is known that every V g k -module is unitarizable and hence Rep + ( V g k ) ≃ Rep( V g k ).Let q = e iπd ( k + h ∨ ) . Then the quantum group category ˜ O q ( g ) is a tensor C ∗ -category by[128, 133]. It follows from the Finkelberg’s equivalence ˜ O q ( g ) ≃ Rep( V g k ) that A ( V g k )admits the structure of a Ω-involutive weak quasi-Hopf C*-algebra and the conclusionfollows from Theorem 17.7. (cid:3) Remark 17.9.
Theorem 17.8 has been recently proved by B. Gui in the special cases g = sl N , N ≥ g = so N , N ≥ + ( V ) admits a tensor C ∗ -structure. Theorem 17.10.
Let V be a unitary vertex operator algebra satisfying assumptions (a),(b), (c) and such that the forgetful functor : Rep + ( V ) → Rep( V ) is a linear equivalenceand assume that the functor F V : Rep( V ) → Vec satisfies the weak dimension conditionin Remark 17.3. Assume that
Rep( V ) is tensor equivalent to a tensor C ∗ -category. Then Rep + ( V ) admits a structure of tensor C ∗ -category with unitary braided symmetry such thatthe forgetful functor : Rep + ( V ) → Rep( V ) is a braided tensor equivalence.Proof. (cid:3) We now give some examples of applications of Theorem 17.10.
Example 17.11.
Let L be an even positive definite lattice and let V L be the correspondingunitary VOA. It satisfies assumptions (a), (b), (c). It follows from [29, Th. 4.12] theforgetful functor Rep + ( V ) → Rep( V ) is a linear equivalence. The fusion ring of Rep( V L )is isomorphic to the finite abelian group L ∗ /L , where L ∗ is the dual lattice of L . For anirreducible V L -module M [ x ] , with equivalence class corresponding to [ x ] ∈ L ∗ /L we have D ( M [ x ] ) = N [ x ] , where N [ x ] is the number of elements of L ∗ in the equivalence class [ x ]having minimal norm, see e.g. [48]. In some cases, e.g. square lattices, one can easily checkthat D is a weak dimension function. The irreducible objects of Rep( V L ) are all invertibletheir equivalence classes form a finite abelian group G ≃ L ∗ /L . It follows that Rep( V L ) is EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 87 tensor equivalent to Vec ωG for some 3-cocycle ω ∈ Z ( G, T ), where Vec ωG is the category of G -graded finite dimensional vector spaces with associators twisted by ω , see [37]. Vec ωG istensor equivalent to the tensor C ∗ -category Hilb ωG of G -graded finite dimensional Hilbertspaces with associators twisted by ω and hence, if V L satisfies the weak dimension property,Rep + ( V L ) admits a structure of a tensor C ∗ -category with unitary braiding symmetrymaking the forgetful functor : Rep + ( V L ) → Rep( V L ) into a braided tensor equivalence. Example 17.12.
Let V be a unitary VOA satisfying assumptions (a), (b), (c) and assumethat V is holomorphic i.e. that Rep( V ) is equivalent to Vec. Let G be a finite subgroup ofthe unitary automorphism group of V and let V G be the corresponding orbifold unitary subVOA. It is conjectured that always V G satisfies (a), (b), (c) and that Rep( V G ) is braidedtensor equivalent to Rep( D ω ( G )) ≃ Z (Vec ωG ), for some 3-cocycle ω ∈ Z ( G, T ). Here, D ω ( G ) the twisted quantum double quasi-Hopf algebra introduced in [27] and Z (Vec ωG ) isthe center of Vec ωG , [37]. This conjecture is known to be true in various cases, see e.g.[18, 30, 81, 95]. Assume now that the above conjecture is true for a given V and G andalso assume that every irreducible V G -module is unitarizable. Since Rep( D ω ( G )) is tensorequivalent to a tensor C ∗ -category then, if V G satisfies the weak dimension property,Rep + ( V G ) admits a structure of a tensor C ∗ -category with unitary braiding symmetrymaking the forgetful functor : Rep + ( V G ) → Rep( V G ) into a braided tensor equivalence.Let us know consider an explict example. Let Λ be the Leech lattice, the even unimodularlattice of rank 24 with trivial root system, and let let V Λ be the corresponding lattice VOA.Since Λ = Λ ∗ , V Λ is holomorphic. V Λ as special automorphism of order two which can easilyseen to be unitary, see [29, Sec. 4.4] where this automorphim is denoted by θ . As usual wedenote by V +Λ the corresponding unitary fixed point subalgebra. V +Λ satisfies (a), (b) and(c). Moreover, up to equivalence it has exactly four irreducible modules V +Λ , V − Λ , ( V T Λ ) + and ( V T Λ ) − which are all invertible and unitarizable [29, 30]. Hence the equivalence classesof irreducibles form an abelian group of order 4 which in fact is isomorphic to Z × Z ,see e.g. [30, Prop.5.6]. Arguing as before can conclude that Rep( V +Λ ) is tensor equivalentto a tensor C ∗ -category. The characters (graded dimensions) of the irreducible modulesof V +Λ are known, see [42, Sec. 10.5] and [115, Prop. 2.5] and from them one can easilycompute the function M D ( M ) = dim F V +Λ ( M ) and we find D ( V +Λ ) = 1, D ( V − Λ ) = 24, D (cid:0) ( V T Λ ) + (cid:1) = 2 and D (cid:0) ( V T Λ ) − (cid:1) = 24 · . It follows that V +Λ has the weak dimensionproperty and hence , by Theorem 17.10, Rep + ( V +Λ ) admits a structure of tensor C ∗ -categorywith unitary braided symmetry such that the forgetful functor: Rep + ( V +Λ ) → Rep( V +Λ )is a braided tensor equivalence. With this structure Rep + ( V ) is a modular tensor C ∗ -category because Rep( V +Λ ) is modular. The modular T matrix of Rep + ( V ) can also becomputed from the characters and it is given by the diagonal matrix with diagonal entries1 , , , −
1. By [113] there is, up to equivalence, a unique unitary fusion category withfusion rules Z × Z , the above T matrix and topological central charge 24 mod 8 andit is realized by the representation category of the quantum double D ( Z ), with trivialtwist ω ∈ H ( Z , T ) ≃ Z . Note that, A ( V +Λ ) and D ( Z ) have equivalent representationcategories but are inequivalent associative algebras. D ( Z ) is commutative while A ( V +Λ )is not. Note also that D ( Z ) is a Hopf algebra while A ( V +Λ ) is a weak quasi-Hopf algebra. In the final part of this section we explain how most of the constructions and resultswe have discussed in the case of rational vertex operator algebras have an analogue inthe case of completely rational conformal nets. These two picture are perhaps related bythe correspondence between unitary vertex operator algebras, conformal nets and theirrepresentations [20, 21, 56, 57].Let A be a completely rational conformal net on S . We denote by Rep( A ) the categoryof (Hilbert space *-) representations of A with finite index. Note that every irreduciblelocally normal representation of A has finite index and hence is an object in Rep( A ).Accordingly the finite index condition is assumed only to rule out infinite Hilbertian directsums. Its known that Rep( A ) is a modular tensor C ∗ -category [74, 75]. Here we brieflydescribe how this structure of modular tensor category is defined. Let I ⊂ S be a givennon-empty non-dense open interval. Then one can define a full C*-subcatefgory Rep I ( A ),Rep( A ) whose objects are the representations localized in I , see e.g. [74, Sec. 3.2.].The objects in Rep I ( A ) gives rise to unital endomorphisms of the type III factor A ( I )and the composition of endomorphisms makes Rep I ( A ) into a strict tensor C ∗ -categorywhich turns out to be modular as a cosequence of the results in [75]. It is known thatEvery representation in Rep( A ) is unitary equivalent to a representation in Rep I ( A ) sothat the embedding I : Rep I ( A ) → Rep( A ) is a unitary equivalence of C*-categories.Accordingly, given any equivalence E : Rep( A ) → Rep I ( A ) with a unitary isomorphism η : E ◦ I → Rep I ( A ) one can transport the modular tensor C ∗ -category structure on Rep( A )and give to E a tensor structure making it into a unitary tensor equivalence. Note thatone can chose E such that E ◦ I = 1 Rep I ( A ) and accordingly η such that η π = 1 π for all π inRep I ( A ). With this choice Rep( A ) turns out to be a strict tensor C ∗ -category.Given a representation π of A with finite index we denote by L π the correspondingconformal Hamiltonian. L π is a self-adjoin operator with pure point-spectrum. In thefollowing we will assume that A satisfies the following(d) For every representation π of A with finite index L π has finite dimensional eigenspaces.Assumption (d) is believed to be always satisfied. It would follow e.g. from [20, Conjec-ture 9.4] or from [74, Conjecture 4.18].We now want to define a conformal net analogue of the functor F V defined at thebeginning of this section. Every representation π of A with finite index on the Hilbertspace H π can be written as a finite direct sum of irreducibles π = M i π i (17.7)and correspondingly a Hilbert space decomposition. H π = M i H π i . (17.8)We denote by h i ≥ L π i and by H π i (0) the corresponding eigenspacewhich is finite dimensional by our previous assumption. We now define a finite dimensional EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 89 closed subspace H (0) ⊂ H by H π (0) := M i H π i (0) . (17.9) H (0) is independent from the choice of the direct sum decomposition in Eq. (17.7). More-over, π ( A ) ′′ H π (0) = H π where π ( A ) ′′ is the von Neumann algebra on H π generated by thealgebras π I ( A ( I )), with I an open non-dense non-empty interval of S .In complete analogy with the VOA case one can define a linear functor F A : Rep( A ) → Hilb by F A ( π ) := H π (0) for any representation with finite index π of A and F A ( T ) := T ↾ H π α (0) for any intertwiner operator T ∈ ( π α , π β ) and it turns out that F A is a faithful *-functor. The algebra A ( A ) := Nat ( F A ) is a finite dimensional C*-algebra and there isa *-equivalence of C*-categories E A : Rep( A ) → Rep + ( A ( V )) which, after compositionwith the forgetful functor : Rep( A ) → Hilb is isomorphic to F A . The algebra A ( A ) is theconformal net analogue of the Zhu’s algebra. The following is the conformal net version ofTheorem 17.1 Theorem 17.13.
Let A be a completely rational conformal net satisfying assumption (d).Assume moreover that π D ( π ) := dim( F A ( π )) , π irreducible, gives a weak dimensionfunction on the modular tensor category Rep( A ) . Then, the algebra A ( A ) admits a struc-ture of a Ω -involutive weak quasi-Hopf C*-algebra with a *-tensor equivalence E A Rep( A ) → Rep + ( A ( A )) which, after composition with the forgetful functor : Rep( A ) → Hilb is tensorisomorphic to F A .Proof. By Theorem 5.9 F A admits a weak quasi-tensor structure and the conclusion followsfrom Theorem 10.5. (cid:3) We conclude this section with a brief comparison of the VOA and the conformal netquasi-Hopf algebras discussed in this section. In [20] a class of unitary simple VOAs called strongly local
VOAs has been introduced and a map V → A V form strongly local VOAsto conformal nets has been defined. It is conjectured in [20] that every simple unitaryvertex operator algebra V i strongly local and that the map V → A V gives a one-to-onecorrespondence between unitary simple VOAs and (irreducible) conformal nets. Moreover,it is conjectured in [74, Conjecture 4.43] that the unitary VOA satisfies assumptions (a),(b) and (c) if and only if A V is completely rational and that, in this case Rep( A V ) andRep( V ) are tensor equivalent, see also [56, 57, 64]. This conjecture appears to be a veryhard and important problem and whose solution for even for a representative class ofexamples is of grat interest. We hope that our work could give some useful hints in thisdirections and we hope to come back to this in future work. Here we limit ourselves togive some hints in the special case of the type A affine vertex operator algebras V sl N k .We now from [20] that, for all N ≥ k ≥ V sl N k is a simple unitary stronglylocal VOA and that the conformal net A V sl N k is isomorphic to the loop group conformal net A SU( N ) k . The latter is known to be completely rational as a consequence of Wassermann’swork [127] and the fusion rules of Rep( A SU( N ) k ) are known to agree with those of Rep( V sl N k ).Actually the two modular tensor categories are known to have the same modular data, i.e. the same modular S and T matrices. Moreover by [21], see also [57], every unitary V sl N k -module M “integrates” to a representation π M of A SU( N ) k on the Hilbert space completion H M of M and the map M π M gives rise to a *-isomorphism of C*-categories E SU( N ) k : Rep + ( V sl N k ) → Rep( A SU( N ) k )and it is straightforward to see that F + V sl N k = F A SU( N ) k ◦ E SU( N ) k . As a consequence we have a canonical isomorphism A ( V sl N k ) ≃ A ( A SU( N ) k ) and we have atensor equivalence Rep + ( V sl N k ) ≃ Rep( A SU( N ) k ) if and only if the weak quasi-Hopf algebrastructures on A ( V sl N k ) and A ( A SU( N ) k ) agree up to a twist.18. Kazhdan-Wenzl theory and equivalence of ribbon s l N,q,ℓ -categories
Let g be a simple complex Lie algebra. We keep the notation fixed in the first paragraphof the previous section for C ( g , q, ℓ ). We recall that the fusion categories C ( g , q, ℓ ) arisingfrom quantum groups at roots of unity are deeply related to fusion categories arising fromchiral CFT on the circle. Let k be a positive integer and let V g k denote the affine VertexOperator Algebra (VOA) of level k with Rep( V g k ) the associated representation category.By results of Huang [61, 62, 63, 64] this is a modular fusion category. In his paper,Finkelberg [39] proved that Rep( V g k ) and C ( g , q, ℓ ) are equivalent as ribbon categories forthe specific roots of unity q = e iπ/ℓ with ℓ = d ( k + ˇ h ). The proof is based on the work ofKazhdan and Lusztig of the early 90s.On the other hand, the approach to CFT via conformal nets [47] provides examples ofmodular fusion categories as well [75]. A general connection from VOA satisfying suitableanalytic conditions to conformal nets has recently been established [20].An important example is the fusion category associated to the loop group conformal netover SU( N ) which is known to have the same fusion rules [127] and modular data (the S and T matrices) as the corresponding affine VOA or quantum group categories. More precisely,the associated Verlinde fusion ring R N,ℓ arises from positive energy representations of thelevel k central extension of the loop group of SU( N ) and also as the Grothendieck ring ofRep( V g k ) or C ( s l N , q, ℓ ) for any q such that q is a primitive root of unity of order ℓ , in thiscase ℓ = k + N see e.g. [5, 114, 3].It is then natural to ask whether there is a classification of ribbon fusion categorieswith Verlinde fusion rules of type A showing in particular ribbon equivalence of the fusioncategories arising from the three different settings. In this section we give a classificationresult independent of Finkelberg equivalence theorem. We shall not assume that ourcategories have a unitary structure, and we replace this condition with the possibly weakerassumption of pseudounitarity in the sense of [38]. In this way our result may be usefulfor the purposes of Sect. 17 for this special case. In that section we construct unitarystructures of the representation category of all the affine vertex operator algebras. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 91 a) Let R N denote the representation ring of SL( N, C ). It is freely generated with basis e λ parameterised by the set of dominant integral weights Λ, so every λ ∈ Λ is a non increasingsequence ( m , . . . , m N − ) of non negative integers.b) For a positive integer ℓ > N , let Λ ℓ be the Weyl alcove recalled at the beginningof the previous section. For g = s l N , Λ ℓ may be described by weights λ ∈ Λ satisfying m ≤ ℓ − N . The Verlinde fusion ring R N,ℓ has a natural basis e λ with λ ∈ Λ ℓ . Thestructure constants are determined by the Verlinde formula [], or via characters of theaffine Weyl group, the Kac-Walton formula []. The fusion ring R N,ℓ may also described asa quotient of R N by a certain ideal, see [25].We set R N, ∞ = R N , so the general notation R N,ℓ will include N + 1 ≤ ℓ ≤ ∞ unless oth-erwise stated (as it will be for example in the main theorem of the section). Furthermore, R N,ℓ will be regarded as a based ring in the sense, e.g., of [102].Note that a semisimple rigid tensor category C with based Grothendieck ring isomorphicto R N,ℓ for ℓ finite is a fusion category.Frobenius-Perron dimensions of basis elements FPdim( X i ) of a commutative based ringwere introduced in [46], and one has FPdim( X i ) >
0. We refer to Sect. 8 in [38] or Chapter4 in [37] for the development of the theory in generality. We shall be interested in the caseof the based Grothendieck ring Gr( C ) of a fusion category C endowed with its natural basisgiven by the equivalence classes of irreducible objects.The main result is that X i → FPdim( X i ) extends uniquely to a homomorphism ofalgebras φ : Gr( C ) → R , and φ is the unique homomorphism such that φ ( X i ) > i ,see Theorem 8.2 and Lemma 8.3 in [38]. The global Frobenius-Perron dimension is definedas FPdim( C ) = P i FPdim( X i ) .The global categorical dimension is in turn defined as the sum of the squared dimensions | X i | of simple objects X i . Squared and global categorical dimensions were introduced andstudied by M¨uger for spherical fusion categories in [92] and extended to general fusioncategories in [38]. It is known that | X i | > C is spherical, | X i | = d ( X i ) , with d the categorical dimension defined via the spherical structure, see Sec. 20. In particular, d ( X i ) is independent of the choice of the spherical structure.A fusion category C is called pseudo-unitary if the global dimension dim( C ) equals theFrobenius-Perron dimension FPdim( C ).The squared dimension of every simple object X i is bounded above by FPdim( X i ) ,hence C is pseudo-unitary if and only if these are all equalities, see Prop. 8.21 in [38]. ByProp. 8.23 of the same paper a pseudo-unitary fusion category admits a unique pivotalstructure, in fact spherical, such that the categorical dimensions of simple objects X i arepositive, or equivalently coincide with the FPdim( X i ).We next specialise to braided fusion categories. In this case, pivotal (spherical) struc-tures are in a natural bijective correspondence with balanced (ribbon) structures for thebraided symmetry, and the correspondence is recalled in Sect. 20. It also follows from theprevious paragraph that a pseudo-unitary braided fusion category admits a unique ribbonstructure inducing positive categorical dimensions. We shall refer to it as the positive ribbon structure. The aim of this section is to show the following result. Theorem 18.1.
Let C and C ′ be pseudo-unitary ribbon fusion categories with ribbon struc-tures θ and θ ′ , assumed positive for N even, and with based Grothendieck rings isomorphicto R N,ℓ with N + 1 ≤ ℓ < ∞ . Let f : Gr( C ) → Gr( C ′ ) be a based ring isomorphism suchthat for each irreducible ρ ∈ C , θ ′ ρ ′ = θ ρ where ρ ′ is an irreducible in C ′ in the class of f [ ρ ] . Then there is an equivalence of ribbon tensor categories F : C → C ′ inducing f . Ifthe categories are unitary, F may be chosen unitary. For N = 2 this result has recently been shown in [10] using Fr¨ohlich-Kerler classification[46]. It follows from Ex. 18.14 that the positivity requirement on the ribbon structurescan not be removed. Moreover it will be clear from the proof how a ribbon structure canbe positive only for a unique braiding. We reformulate the previous result in a form usefulfor applications. Corollary 18.2.
Let C and C ′ be modular fusion categories with positive categorical di-mensions and with Grothendieck rings isomorphic to the Verlinde fusion ring R N,ℓ via anisomorphism compatible with the corresponding T -matrices. Then C and C ′ are equivalentas ribbon tensor categories.Proof. The categories are pseudo-unitary by positivity of the categorical dimensions. Com-patibility of the T -matrices implies compatibility of the ribbon structures. The conclusionfollows from Theorem 18.1. (cid:3) Definition 18.3.
Following [78], a semisimple rigid tensor category C together with anisomorphism of based rings φ C : R N,ℓ → Gr( C ) is called of s l N,ℓ -type. Two s l N,ℓ -typecategories ( φ C , C ) and ( φ C ′ , C ′ ) are equivalent if there is a tensor equivalence E : C → C ′ inducing an isomorphism between the Grothendieck rings compatible with φ C and φ C ′ .The proof of Theorem 18.1 will occupy the rest of this section and it is based on Kazhdan-Wenzl theory [78]. To summarize, Kazhdan-Wenzl theory gives a classification of s l N,ℓ -typetensor categories in terms of the categories arising from quantum groups both for genericor root of unity values of the deformation parameter q , and a 3-cocycle on the dual ofthe center of SU( N ) which modifies the natural associator. We start recalling the mainresult. We shall then show that the positive ribbon structure completely determines theribbon tensor category under our assumptions. The most delicate part of our analysisis a characterization of braided pseudo-unitary s l N,ℓ -type fusion categories among general s l N,ℓ -type categories, stated as Cor. 18.13, and relies on the theory of quasitriangularw-Hopf algebras developed in the paper. We also give a parameterisation of the braidedsymmetries and a classification of their ribbon structures that is useful in our proof. Werecall for completeness that a characterization of braided s l N, ∞ -type categories has beenmade explicit in [99], see also [70], and a classification of the braided symmetries may befound in [105]. ℓ = N + 1 . The based ring R N,N +1 identifies with ZZ N , with basis Z N the cyclic group of order N . Hence a s l N,N +1 -type fusion category C ispointed over Z N . By Prop 4.1 in [78], see also Example 5.12 and references therein, Vec ω Z N exhaust the s l N,N +1 -categories, which are classified by ω ∈ H ( Z N , T ). A general braided EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 93 pointed fusion category over the finite abelian group G of equivalence classes of irreducibleobjects determines a quadratic form on G via q ( g ) = c ( γ, γ ), where g is the class of γ .The pair ( G, q ) determines C as a braided tensor category by Theorem 8.4.9 in [37]. ByRemark 4.13 in [97], if θ is the ribbon structure associated to a braided symmetry c and aspherical structure in a fusion category then on every object X , θ X = Tr X ⊗ c ( X, X )). Ina pointed fusion category c ( X, X ) is a scalar and d ( X ) = ± X is irreducible, and hence d ( X ) = 1 under the positivity requirement, and therefore θ X = c ( X, X ). Hence q ( g ) = θ γ .In other words the datum ( G, q ) is equivalent to that of the fusion rules and the positiveribbon structure. The result applies in particular to s l N,N +1 -type categories and the proofis complete in this case.As remarked in Ex. 15.1 these categories are unitary in a natural way, so the pseudo-unitarity assumption holds automatically. Examples of s l N, ∞ -type categories are the representationcategories of quantum s l N -groups for generic values of the deformation parameter. Specif-ically, the quantum group of [44] was originally considered in [78]. Being a quantizationof a Hopf algebra of functions, the category is described by corepresentations. In thesetting of tensor C ∗ -categories, it is natural to consider the category of unitary corepre-sentations of Woronowicz SU q ( N ) group, where q is real, this is e.g. the starting point of[103, 105, 99, 70]. We refer to [100] for details on the natural tensor C ∗ -structure. In a moregeneral framework where a C ∗ -structure is not assumed, one may consider the categoryof representations of the Drinfeld-Jimbo quantum group U q ( s l N ) for q a non-zero complexnumber, not a nontrivial root of unity. By representations we understand those which canbe obtained as direct sums of subrepresentations of tensor products of Weyl modules. If q is positive, SU q ( N ) and U q ( s l N ) induce equivalent tensor categories, see [100]. To unifywith the examples C ( s l N , q ) at roots of unity, we shall adopt Drinfeld-Jimbo framework.In the following we assume ℓ > N + 1. Then s l N,ℓ -category is determined up to tensorequivalence by two invariants, q C and τ C , a pair of nonzero complex numbers, unique up topassing to the pair with reciprocal values, which determines the tensor category, togetherwith the fixed isomorphism φ , up to equivalence. These invariants are defined, and relatedto each other, as follows.Let X ∈ C be an object in the class of the image of (1 , , . . . ,
0) under φ C . The tensorproduct of X with any irreducible is multiplicity free, and the fusion rules can be foundin [78]. Let a ∈ ( X , X ) be the idempotent onto the subobject (1 , , , . . . , q C (unique up to passing to the inverse) such that T := q C ( I − a ) − a ∈ ( X , X ) gives rise via the usual construction T i = 1 i − ⊗ T ⊗ n − i − to arepresentation of the braid group π n : B n → ( X n , X n ). If g , . . . , g n − are the generators of B n , thus satisfying the presentation relations g i g i +1 g i = g i +1 g i g i +1 , π n takes g i → T i . In ourformulas, for simplicity, we are assuming that the category is strict. This representationfactors through the defining relations ( g i − q C )( g i + 1) = 0, i = 1 , . . . , n −
1, of the Heckealgebra H n ( q C ) since a is an idempotent. Thus we have representations of the Hecke algebras denoted with the same symbol, π + n : H n ( q C ) → ( X n , X n )compatible with the tensor structure. The ambiguity in the choice of q C also gives π ′ n : H n ( q C − ) → ( X n , X n ), which may equivalently be thought of as another Hecke algebrarepresentation on the same parameter π − n : H n ( q C ) → ( X n , X n ) , the opposite , or dual representation via π − n := π ′ n β = π = n α using the canonical isomorphism β : H n ( q ) → H n ( q − ) which relates the corresponding canonical generators via g i → − qh i ,and α : g i ∈ H n ( q ) → q − − g i ∈ H n ( q ).Let C q,N,ℓ denote C ( s l N , q, ℓ ) for q a primitive root of unity of order ℓ , for ℓ < ∞ and C q,N, ∞ the category Rep( U q ( s l N )) for q not a non-trivial root of unity. Note that C q,N,ℓ does not change, up to tensor equivalence, under the passage from q to q − . This maybe seen as follows. For ℓ = ∞ there is an isomorphism from the quantum group U q ( g ),to U q − ( g ) given by E i → K i F i , F i → E i K − i , K i → K i . For ℓ < ∞ we may use ananalogous isomorphism for U x ( g ), where x is now an indeterminate, and the quantumgroup is regarded over C ( x ), (see [114], with our x corresponding to q ), and taking intoaccount Lusztig’s specialization of U x ( g ) to U q ( g ) for q is a complex primitive root of unity.For details see e.g. in Sect. 9.3, and 11.2 in [22] (for q of odd order) and [114].The category C q,N,ℓ becomes an s l N,ℓ -type category as follows. Set X = X q , the natural N -dimensional representation of U q ( s l N ), and φ q : R N,ℓ → Gr( C q,N,ℓ ) the natural identifi-cation. We realize T as the element − σ defined in (4 .
13) of [132], with q in place of µ andconsider the associated Hecke algebra representations π n .For a general s l N,ℓ -category, it turns out that q C is a primitive root of unity of order ℓ for ℓ finite, and is not a nontrivial root of unity for ℓ infinite. In the first case, H n ( q C ) is notsemisimple for large values of n . In both cases, the kernels of π + n and π − n are completelydetermined by the fusion rules, and the two representations are distinguished by the valuetaken by a certain scalar invariant µ C , see Theorem 4.1 in [78], which corresponds to thevalue of a categorical left inverse of X on T , in the sense of [82] in the Hecke category.The second invariant, called the twist of the category, is given by τ C = p ⊗ X ◦ T ,N ◦ X ⊗ ν ∈ ( X, X ) ≃ C , where ν ∈ ( ι, X N ) and p ∈ ( X N , ι ) satisfy p ◦ ν = 1 and T ,N = T N . . . T is an Hecke algebra element in the representations π n exchanging the first factor in a tensorproduct of N + 1 objects with the following N factors. More precisely, if the category isnot strict, X N = (( X ⊗ X ) ⊗ X ) . . . and we need to use associativity morphisms in defining τ C .Given C , with associativity morphisms α , and given a N -th root of unity w , we mayconsider a new tensor category, C w with the same representation ring, the same structureas C except for the associativity morphisms, which are modified as follows, α wX λ ,X µ ,X ν := w γ ( | λ | , | µ | ) | ν | α X λ ,X µ ,X ν , (18.1)for λ , µ , ν ∈ Λ (or in Λ ℓ accordingly), where γ is the function γ ( a, b ) = [ a + bN ] − [ aN ] − [ bN ]and | λ | = m + · · · + m N − . EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 95
It is easy to see that q C does not change when passing to a twisted category. This is notthe case for τ C , which does change and in fact determines the root of unity w defining thetwist. Indeed, starting with a given C as before, if X is the conjugate of X naturally realizedas a subobject of X N − , we have ν ∈ ( ι, X ⊗ X ). Taking into account the associativitymorphisms, it follows that τ C is the composite ( X = X ) X −→ X ( XX ) α − −→ ( X X ) X T ,N − ⊗ −→ ( XX ) X (18.2) α −→ X ( X X ) ⊗ T , −→ X ( XX ) α − −→ ( XX ) X −→ X . (18.3)Passing from C to C w gives rise to a modification in the computation of the correspondinginvariant only on the associativity morphisms. More precisely, the second part, (18.3), doesnot change, by centrality of the deforming factor in α w , see (18.1), while (18.2) changes bya factor w − . This follows from a simple computation, since X corresponds to (1 , . . . , τ C w = w − τ C .The following theorem is due to Kazhdan and Wenzl [78]. For clarity sake we include aproof. Theorem 18.4.
Let C be a s l N,ℓ -type tensor category with N + 1 < ℓ ≤ ∞ , φ C : R N,ℓ → Gr( C ) an isomorphism and let X , q C and τ C be defined as above. Then there is a N -th rootof unity w such that τ C = ( − N w − q N − , where q is a complex square root of q C . The pair ( q C , τ C ) is unique up to the pair with reciprocal values and determines the pair ( C , φ C ) upto equivalence. Furthermore, there is an equivalence of ( C , φ C ) with ( C wq,N,ℓ , φ q ).Proof. Kazhdan-Wenzl left inverse µ C takes the value stated in Theorem 4.1 in [78] on T . It follows that the representation π n of the Hecke algebra is quasi equivalent to thatarising from the quantum group in C q,N,ℓ . In the generic case, a computation of τ C asin the statement may be found e.g. in Lemma 8.1 of [105] with T i corresponding to − g i there, based on a computation of the left inverse on the generator T for C = C q,N,ℓ andthe mentioned Hecke algebra representation of the quantum group, see Prop. 4.1 andTheorem 3.3 (a) in [103], with N and µ C in turn corresponding to d and λ − d there. Seealso [70]. In the root of unity case, we may argue in the same way, using now Theorem3.3 (b) [103] and replacing S with the morphism still denoted S of the appendix of [23],and derive in a similar way an N -th root of unity w such that τ C takes the stated value.We then conclude following [78]: up to passing to C w − , we may assume with no loss ofgenerality that τ C = ( − N q N − , by (18.2), and (18.3). We have thus reduced the valuesof the invariant q C , τ C to those it would take on C q,N,ℓ . It is easy to see that this valueof τ C in the twisted category means that the element ν ∈ ( ι, X N ) of [78] and the Heckealgebra representations together satisfy the setting of section 6 in [103], that is equations(6 . . C q,N,ℓ has been exhibited for q realtaking X to X q , thus compatible with φ and φ q . More precisely, braided symmetries areconstructed from certain normalizations of the Hecke algebra generator which is necessaryto match T with the R -matrix of the quantum group in the representation X . There isminimal change for other generic values of q . For the root of unity case, we may arguesimilarly, using the information and analogous equations in the appendix of [23] again. (cid:3) It will be useful for us to specialize Kazhdan-Wenzl theory to the untwisted tensorcategories. In the following result, ≃ denotes an equivalence between pairs ( C wq,N,ℓ , φ q ). Corollary 18.5.
Let q ∈ C × be either not a non trivial root of unity or such that q isa primitive root of unity of order ℓ > N + 1 , and let q ′ ∈ C × be another complex numberwith the same property. Then:For N even, a) C q,N,ℓ ≃ C q ′ ,N,ℓ if and only if q ′ = q or q ′ = q ; b) C − q,N,ℓ ≃ C − q,N,ℓ .For N odd, C q,N,ℓ ≃ C q ′ ,N,ℓ if and only if q ′ = ± q , q ′ = ± q . s l N,ℓ -type categories.
Since the work of [44, 132] andthe theory of universal R -matrix of Drinfeld, see e.g. [22], it has been known that U q ( g )gives rise to braided tensor categories. For the case of C ( s l N , q ) see e.g. [114]. There isa simple parameterisation of all the possible braided symmetries of C q,N,ℓ . We start withthe two canonical braided symmetries, ε + and its opposite ε − derived from the R -matrixof the quantum group and its opposite, R − , respectively, see also remark 18.7. Proposition 18.6.
Let z and z ′ vary among the N -th roots of unity. Then for N + 1 <ℓ ≤ ∞ there are N braided symmetries, ε + z and ε − z ′ of C q,N,ℓ uniquely determined by ε + z ( X, X ) = zε + ( X, X ) , ε − z ′ ( X, X ) = z ′ ε − ( X, X ) . Furthermore, this is a complete list.Proof.
Since C q,N,ℓ admits X as a generating object, any braided symmetry c is determinedby c ( X, X ) thanks to (3.7), (3.8), (3.9) By the fusion rules of X , a suitable normalizationof c ( X, X ) will induce a representation of a Hecke algebra. By Kazhdan-Wenzl theory,the eigenvalue of the properly normalized c ( X, X ) corresponding to I − a can only be q ± C ,so that c ( X, X ) is a scalar multiple of ε + ( X, X ) or ε − ( X, X ). By naturality of c ( X, X )on the morphism ν ∈ ( ι, X N ), the scalar is a N -th root of unity. Conversely, for any N -th root of unity z , the modified morphisms c z ( X n , X m ) = z nm c ( X n , X m ) still satisfythe same relations and also the naturality property on the full subcategory with objectstensor powers of X , and hence everywhere, as a consequence of ( X n , X m ) = 0 if and onlyif n ≡ m (mod N ). We may then apply these considerations to ε + and ε − . (cid:3) Remark 18.7.
The braided symmetries described in the previous proposition are perhapsmore clearly explained by the specialization process of the R -matrix of the quantum group.More precisely, this matrix, at the level of the integral form U † A ′ ( g ) of U x ( g ), with x a formalvariable as in [114] where our x corresponds to q in that paper, depends on a root s oforder L of x via s L = x , where L is the smallest integer such that for any pair of dominantweights λ , µ , L h λ, µ i is an integer. The values of L are listed in table 1 in [114]. Wethen specialize x to a primitive complex root of unity q , and let ℓ ′ be its order and s toa fixed but arbitrary complex L -th root q /L of q . Note that our q /L is not necessarily aprimitive root of unity of order Lℓ ′ as in Sect. 2 in [114], thus our specialization needs to EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 97 be slightly generalized. One has L = N for g = s l N . It follows that X ⊗ X ( R ) correspondsto the operator computed in Sect. 8.3G of [22], where e h corresponds to x and gives riseto our braided symmetry ε − through ε − ( X, X ) = Σ X ⊗ X ( R ). The N possible choices of s = q /N give the symmetries z ′ ε − , and a similar relation holds between the specializationof opposite R -matrix R − and the symmetries zε + .The 2 N braided symmetries of Prop. 18.6 give rise to braided tensor categories ( C q,N,ℓ , ε ± z ).We shall need the following property. Remark 18.8.
Our aim is to show that the identity isomorphism between the represen-tation rings of any two of ( C q,N,ℓ , ε ± z ) can not be induced by a braided tensor equivalence.An explicit proof of this fact between two categories of the kind ( C q,N,ℓ , ε + z ) (or ( C q,N,ℓ , ε − z ))which fixes the generating object X may be found e.g. at page 8 of [105] for q real. Thosearguments extend to a nonzero complex generic q or to the root of unity case with the samemodifications indicated in the proof of Theorem 18.4. Since an isomorphism between twoobjects in a braided tensor category induces a braided tensor equivalence between the fullbraided tensor subcategories they generate, it also follows that there is no braided tensorequivalence which takes the generating object X to an equivalent object, and the conclusionfollows in this case. On the other hand for a pair of the kind ( C q,N,ℓ , ε + z ) and ( C q,N,ℓ , ε − z ′ ),an argument may be found in the proof of Theorem 18.1 relying on the comparison of theribbon structures.We refer the reader also to [15, 16] for further studies on these braided symmetries.Up to a sign change of q = ( q C ) / for N even, an s l N, ∞ -type braided tensor category C is tensor equivalent to some Rep( U q ( s l N )). (The case N = 2 holds without the braidedsymmetry requirement, as it follows from the work of [46], or also from Theorem 18.4, since H ( Z , T ) ≃ Z , see also Cor. 18.5.) For N > U q ( s l N )) which provides a discrete Hopfalgebra. We need to extend this result s l N,ℓ -type categories for ℓ < ∞ . However it is notobvious how to modify the methods of [99] for general q (with q is a primitive root ofunity of order ℓ ) as the categories C ( s l N , q ) are not associated to Hopf algebras. Perhapsthe most natural way to proceed is to restrict to some subclass large enough to hold ourapplications. We shall thus first consider only the roots of unity q = ± e ± iπ/ℓ for a choice ofthe square root of Kazhdan-Wenzl invariant q C . This will enable us to replace the role ofthe discrete Hopf algebra of [99] with the w-Hopf algebra A = A W ( s l N , ℓ ) of Sect. 24, itsquasi-triangular structure developed in Sect. 7 and the notion of 3-coboundary associatorfor w -Hopf algebras, Sect. 6. We shall include a proof since it becomes slightly moretechnical due to non-triviality of the associator of A . Proposition 18.9.
Let ( C , φ C ) be an s l N,ℓ -type tensor category and assume that either ℓ = ∞ or q = q / C = ± e ± iπ/ℓ . Then C admits a braided symmetry if and only if w = 1 for N odd and w = ± for N even.Proof. The case ℓ = ∞ ( q generic) has been considered in [99]. By Kazhdan-Wenzl theoryan s l N,ℓ -type category ( C , φ C ) is equivalent to ( C ( s l N , q, ℓ )) w , φ q ). For the case q = ± e ± iπ/ℓ recall that the w -Hopf algebra A W ( s l N , q, ℓ ) of Sect. 24 has representation category tensorequivalent to C ( s l N , q, ℓ ). Let, as before, ∆ and Φ = 1 ⊗ ∆( P )∆ ⊗ P ), P = ∆( I ), be thenatural coproduct and associator of A . Consider the weak quasi bialgebra A w = ( A, ∆ , Φ w ),with the new associator Φ w = ΦΥ w , where Υ w = Υ ∈ A ⊗ A ⊗ A is the central invertibleelement given by Υ = w γ ( | λ | , | µ | ) | ν | . Let us regard C ( s l N , q, ℓ ) as tensor equivalent to Rep( A )and therefore C ( s l N , q, ℓ ) w to Rep( A w ). Let R q denote the R -matrix of A , hence by Prop.7.4 ∆ ⊗ R q ) = Φ ( R q ) ( R q ) Φ , ⊗ ∆( R q ) = Φ − ( R q ) ( R q ) Φ − . If we assume that C is braided then so is Rep( A w ), hence by duality A w is quasi-triangular.Let R be the corresponding R -matrix. Thus R satisfies equations (7.2)–(7.5) with respectto Φ w . Since Υ = Υ , taking also into account the computations in the proof of Prop.7.4, equations (7.4)–(7.5) become∆ ⊗ R ) = Φ R R ΦΥ , ⊗ ∆( R ) = Υ − Φ − R R Φ − . We consider the twist F = R − q R , cf. (7.15), which satisfies ∆ F = ∆ and I ⊗ F ⊗ ∆( F ) = [ I ⊗ R q ⊗ ∆( R q )] − I ⊗ R ⊗ ∆( R ) ,F ⊗ ⊗ F ) = [∆ op ⊗ R q ) R q ⊗ − ∆ op ⊗ R ) R ⊗ . We set, as before, P = a ⊗ b , ∆( a ) = a ⊗ a , ∆( b ) = b ⊗ b and compute I ⊗ R q ⊗ ∆( R q )[∆ op ⊗ R q ) R q ⊗ − =( R q ) Φ − ( R q ) ( R q ) Φ − ( R q ) − Φ − ( R q ) − ( R q ) − Φ − =( R q ) Φ − ( R q ) ( R q ) ∆ ⊗ P )1 ⊗ ∆( P )∆ ⊗ P )( R q ) − b ⊗ a ⊗ b ( R q ) − ( R q ) − Φ − =( R q ) Φ − ( R q ) ( a ⊗ a ⊗ b )( b ⊗ a ⊗ b )( R q ) − ( R q ) − Φ − =( R q ) ( b ⊗ a ⊗ a )( R q ) (1 ⊗ ∆( P )∆ ⊗ P )1 ⊗ ∆( P )) ( R q ) − ( R q ) − Φ − =( R q ) ( b ⊗ a ⊗ a )( b ⊗ a ⊗ b )( R q ) − Φ − =( R q ) (∆ ⊗ P )1 ⊗ ∆( P )∆ ⊗ P )) ( R q ) − ( b ⊗ b ⊗ a ) =( b ⊗ a ⊗ a )( b ⊗ b ⊗ a ) = Φ − . Hence, using centrality of Υ,[ I ⊗ F ⊗ ∆( F )] − F ⊗ ⊗ F ) =[ I ⊗ R ⊗ ∆( R )] − I ⊗ R q ⊗ ∆( R q )[∆ op ⊗ R q ) R q ⊗ − ∆ op ⊗ R ) R ⊗ Φ R − R − Φ R − Φ − Φ R R Φ R Υ =Υ Φ R − R − Φ R Φ R Υ = ΦΥ Υ , we have omitted the computations leading to the last equality, as they are very similar tothe previous ones. Hence ΦΥ Υ satisfies (6.1), and one may similarly establish validity(6.2), thus ΦΥ
Υ is a 3-coboundary associator which may be twisted to Φ by F by Prop.6.13. On the other hand as observed in [99] (Υ w ) Υ w is cohomologous to Υ w on the dualof the center of SU( N ), and therefore we find a tensor equivalence between C ( s l N , q, ℓ ) and C ( s l N , q, ℓ ) w which identifies the generating representations, and hence is compatible with EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 99 the chosen isomorphisms with R N,ℓ . From Kazhdan-Wenzl classification we derive w = 1and we finally apply Cor. 18.5. (cid:3) We note that a result closely related to Prop. 18.9 has also been obtained in [11] in the C ∗ -case with different methods. If a given ribbon structure for the braided symmetry of a fusioncategory C induces a spherical structure making the categorical dimensions of the simpleobjects positive then C is pseudo-unitary. It follows that C ( s l N , q, ℓ ) is pseudo-unitary for q = ± e ± iπ/ℓ and N + 1 < ℓ < ∞ with respect to the natural ribbon structure, by [2] (andin fact unitary by [128, 134]). In this subsection we prove that these fusion categories maybe intrinsically characterized among general fusion s l N,ℓ -type categories by the property ofbeing both braided and pseudo-unitary.
Proposition 18.10.
Let q ∈ C be such that q is a non-trivial root of unity of order ℓ > N + 1 . Then C ( s l N , q, ℓ ) is pseudo-unitary if and only if q = ± e ± iπ/ℓ .Proof. Our proof follows that of an analogous result for the Lie type B given in Theorem3.8 in [112], with a slight modification due to the non-uniqueness of the spherical structuresfor N even in our case, see the following Prop. 18.11. More in detail, we write q = ± q z ,with q z = e iπz/ℓ and z an integer with 1 ≤ z ≤ ℓ − z, ℓ ) = 1. Let X bethe object of C ( s l N , q, ℓ ) corresponding to the fundamental representation and assume N = 2 k even. Up to a sign, the categorical dimension d ( X ) with respect to any sphericalstructure equals d q z ( X ) := q zN − + q zN − + · · · + q z − ( N − = 2 P kj =1 cos((2 j − πz/ℓ ).Furthermore, FPdim( X ) = d q ( X ′ ) where X ′ is a corresponding object in the category C ( s l N , q ) for q = e iπ/ℓ , since these two categories have isomorphic representation ringswith an isomorphism identifying X to X ′ and we know that d q takes positive values onthe irreducibles. We claim that d q z ( X ) < d q ( X ′ ) for z = 1. Thus if d q z ( X ) > | d ( X ) | = d q z ( X ) which then can equal FPdim( X ) only if q = ± e iπ/ℓ . If d q z ( X ) < | d ( X ) | = − d q z ( X ) = 2 P kj =1 cos((2 j − π ( ℓ − z ) /ℓ ). Since ℓ − z satisfies the sameproperties as z , pseudo-unitarity again implies ℓ − z = 1 hence q = ± e − iπ/ℓ . To show theclaim, observe that the set S of points q j − = e i (2 j − π/ℓ , j = 1 , . . . k all lie in the uppersemicircle. Furthermore the conditions gcd( z, ℓ ) = 1 and ℓ ≥ N + 2 imply ℓ ∤ (2 j − z .In particular q z j − = 1 for all j . Assume that z is such that the subset S of { q ± (2 j − z } contained in the upper semicircle differs from S . The first point in the natural order of thesemicircle is q . Furthermore two adjacent points of S correspond to arcs whose distanceis at least 2 π/ℓ . Therefore there must be an element of S in between unless they both lieafter the last q N − . Since cos is an even function, it follows that d q z ( X ) may be computedconsidering elements of S , and we have d q z ( X ) < d q ( X ′ ) by the above remarks. We areleft to show that for z = 1, S = S . For this we may apply arguments analogous to thoseof the last part of the mentioned theorem of [112].In the case where N is odd the proof is simplified by the fact that C ( s l N , q, ℓ ) admitsa unique spherical structure, so d ( X ) is uniquely determined. We may thus complete the
00 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI proof with argument similar to the even case, taking into account the additional informa-tion that d ( X ) = q N − z + · · · + q − ( N − z = − P kj =0 cos(2 jπz/ℓ ) where N = 2 k + 1. (cid:3) For completeness we recall from Example 15.1 that C ( s l N , q, ℓ ) are always unitary if q is a primitive root of unity of order ℓ = N + 1. Here below we remark about classificationof spherical structures on C q,N,ℓ . Proposition 18.11.
For N odd, C q,N,ℓ has a unique spherical structure, for N even it hastwo.Proof. In a fusion category C spherical structures are parameterised by the group ofmonoidal natural transformations from the identity functor to itself and taking values ± C admits a simple generating object X , any such natural transformation η is de-termined by the value it takes on X as follows. If η X = λ X then on any tensor power, η X r = λ r X r by monoidality. It follows from naturality and complete reducibility that thevalues that η takes on the simple summands of X r also coincide with λ r . Hence if λ = 1then η is the identity natural transformation, while if λ = − η takes value 1 ( − X . In our specific case, if X = X q we must have λ N = 1 since the tensor unit is a subobject of X Nq . Hence for N odd theconclusion follows. For N even, the specific fusion rules of an s l N -type tensor categoryshow that any odd tensor power of X q is disjoint from an even tensor power. This impliesexistence of a monoidal natural transformation η ∈ (1 ,
1) taking these values. (cid:3)
The next step is that of characterizing general pseudo-unitary s l N,ℓ -type fusion categoriesfor ℓ > N + 1. To do this, we regard the relationship between C ( s l N , q, ℓ ) w , and C ( s l N , q, ℓ )as an example of a general construction described in [12] of a new fusion category C ω fromfrom a given one C and a T -valued 3-cocyle ω on the chain group Ch( C ), and we studyinvariance of pseudo-unitarity under ω in this framework.Let C be a semisimple monoidal category with associativity morphisms α . The chaingroup Ch( C ) introduced in [9, 53] is defined as follows. Consider a complete family Irr( C ) = { ρ α , α ∈ A } of simple objects of C endowed with the smallest equivalence relation ≃ makingall the irreducible subobjects ρ γ appearing in the decomposition of ρ α ⊗ ρ β for fixed α , β ∈ A , equivalent. Then Ch( C ) = Irr( C ) / ≃ is a group with [ ρ α ][ ρ β ] = [ ρ γ ]. The trivialelement is the class of the tensor unit, and [ ρ α ] − = [ ρ α ]. This is an interesting group.For example, it identifies naturally with the dual of the centre of the compact group G for C = Rep( G ) [93]. Furthermore, the group of nonzero C -valued homomorphisms on Ch( C )identifies with the group of natural monoidal transformations of the identity functor on C [53, 12]. Finally, for modular categories, Ch( C ) identifies with the dual of the (abelian)group of invertible elements of C [53].The chain group induces a grading on C , in the sense that there are full subcategories C g indexed by elements of g ∈ Ch( C ) such that every object ρ ∈ C decomposes uniquely upto isomorphism into a direct sum of objects ρ g ∈ C g and with the property that for g = h ,objects of C g are disjoint from objects of C h . The group structure of Ch( C ) implies that EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 101 the grading is compatible with the tensor structure: ι ∈ C e and ρ ⊗ σ ∈ C gh for ρ ∈ C g , σ ∈ C h .We consider C ω , the monoidal category with the same structure as C except for thethe associativity morphisms, which are given by α ωρ,σ,τ = ω ( g, h, k ) α ρ,σ,τ , where [ ρ ] = g ,[ σ ] = h , [ τ ] = k . Note that C ω may be regarded as a special case of a categorical analogueof Prop. 5.11. In other words, C and C ω have isomorphic Grothendieck rings and chaingroups, and, in the framework of fusion categories, they have the same Frobenius-Perrondimension function. We denote by d C ( ρ ) and d C ω ( ρ ) the categorical dimensions of an object ρ considered in C or C ω respectively with respect to preassigned spherical structures. Proposition 18.12.
Let C be a fusion category and ω ∈ Z (Ch( C ); T ) normalized. If D is the right duality functor of C associated to the right duality ( ρ ∨ , b ρ , d ρ ) and η ∈ (1 , D ) is a pivotal (spherical) structure then ( ρ ∨ , b ω , d ω ) is a right duality for C ω where b ωρ = b ρ , d ωρ = d ρ ω − ( g, g − , g ) , with ρ simple and [ ρ ] = g . Furthermore η ω = η is a pivotal(spherical) structure for the associated right duality functor D ω . In particular, if η isspherical under the correspondence ( η, D ) → ( η ω , D ω ) we have d C ( ρ ) = d C ω ( ρ ) for everyobject ρ . Furthermore, C is pseudo-unitary if and only if so is C ω .Proof. Let ρ be an object of C g and let ( b, d ), ( b ′ , d ′ ) solve the right and left dualityequations respectively for ρ in C in the sense of (3.1)–(3.4) with ρ ∨ = ∨ ρ . Then a solutionof the corresponding equations in C ω is given by ( b ω , d ω ), ( b ′ ω , d ′ ω ) where b ω = b , d ω = dω − ( g, g − , g ), b ′ ω = b ′ ω ( g, g − , g ), d ′ ω = d ′ . To verify the duality relations it is useful torecall the equality ω ( g, g − , g ) = ω ( g − , g, g − ) − which follows from the 3-cocycle equationfor ω .We now start with a right duality ( ρ ∨ , b ρ , d ρ ) in C and recall that the associated rightduality functor D was defined in (3.6). The right duality functor D ω of C ω associatedwith the solution ( ρ ∨ , b ωρ , d ωρ ) of the previous paragraph acts as D on objects, while onmorphisms T ∈ ( ρ, σ ) with σ ∈ C h we have D ω ( T ) = ω − ( h, h − , h ) D ( T ). Let η ∈ (1 , D )be a pivotal structure. Consider the left duality ( ρ ∨ , b ′ ρ , d ′ ρ ) defined by (20.3) with η inplace of u . It follows that C ω has left duality ( ρ ∨ , b ′ ωρ , d ′ ωρ ). The natural transformation η ω in C ω defined by (20.4) with d ′ ω and b ω in place of d ′ and b takes the same values as η .Furthermore the natural transformation say F ρ,σ in C making D into a tensor functor isalso natural in C ω and makes D ω into a tensor functor. Indeed, it is easy to see that D ω acts as D on α ρ,σ,τ if ρ , σ , τ are homogeneous, and therefore in general. It follows thatvalidity of Def. 2.5 for D implies validity for D ω by linearity. Hence η ω is monoidal byDef. 2.4, and therefore is a pivotal structure in C ω which is spherical if so was η . Theformulas also show that d C ( ρ ) = db ′ = d ω b ′ ω = d C ω ( ρ ) with respect to these structures.Since C and C ω have the same global FPdim, the last assertion is also clear. (cid:3) Corollary 18.13.
Among the s l N,ℓ -type tensor categories ( C , φ C ) with N + 1 < ℓ < ∞ only those equivalent to some ( C ( s l N , q, ℓ )) w , φ q ) (( C ( s l N , q, ℓ ) , φ q ) resp. ) with q = e iπ/ℓ for N odd and q = ± e iπ/ℓ for N even are pseudo-unitary (pseudo-unitary and braided resp.).Proof. This follows immediately from Propositions 18.10, 18.12, 18.9, Cor. 18.5. (cid:3)
02 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI ℓ > N + 1 . Proof.
Let us fix an isomorphism of based rings φ C : R N,ℓ → Gr( C ). Then ( C , φ C ) is equiv-alent to ( C ( s l N , q, ℓ ) , φ q ) with q = e iπ/ℓ for N odd and precisely to one of ( C ( s l N , q, ℓ ) , φ q )where q takes the values q = ± e iπ/ℓ for N even, according to an equivalence E in-ducing φ C , by Cor. 18.13. A similar conclusion holds for ( C ′ , φ C ′ ) for any choice of φ C ′ : R N,ℓ → Gr( C ′ ). We fix φ C ′ = f ◦ φ C , and denote by E ′ the corresponding equiv-alence with ( C ( s l N , q ′ , ℓ ) , φ q ′ ). Using the based ring isomorphisms induced by E and E ′ between the Grothendieck rings, their compatibility with φ C and φ C ′ , we find an isomor-phism g : Gr( C ( s l N , q, ℓ )) → Gr( C ( s l N , q ′ , ℓ )) which identifies the classes of the respectivegenerating representations X q and X q ′ . Let us now take into consideration the braidedsymmetries, say c and c ′ of C and C ′ respectively, and their ribbon structures, identifiedwith analogous structures in the quantum group categories via the equivalences and de-noted in the same way. For C ( s l N , q, ℓ ) we can only have c = zε + or c = z ′ ε − by Prop. 18.6,where z and z ′ have the same meaning. Taking into consideration Remark 18.7, we identifyeach of the 2 N possible braided symmetries with one derived from the R -matrix R or theopposite R − , subject to a choice of a complex N -th root q /N . Then C ( s l N , q, ℓ ) becomesa ribbon category with positive ribbon structure ˜ θ λ = q ±h λ,λ +2 ρ i , where h , i is a symmetricinvariant bilinear form of s l N such that h α, α i = 2 for (short) roots, the plus or minus signare determined by the choice of R or R − , see [22]. On the other hand C ( s l N , q, ℓ ) alsohas the positive ribbon structure θ λ , hence θ λ = ˜ θ λ by uniqueness of the positive ribbonstructures recalled before the statement of Theorem 18.1. Assuming that θ corresponds tothe plus sign we have that θ X q = q N − N (more details on this formula may be found in theproof of the following proposition). We claim that we may assume that θ ′ corresponds toa plus sign as well. Hence we similarly have θ X q ′ = q ′ N − N . If N is odd we have alreadysettled q = q ′ and our assumption θ X q = θ X q ′ shows that we are taking the same N th rootof q , and therefore we have a braided, in fact ribbon, tensor equivalence. If N is even then N − θ NX q = θ NX q ′ we may exclude that q and q ′ have opposite signs. Itfollows again that the two N th roots of q are the same and we get the same conclusion.We finally show the claim. If on the contrary we had an opposite symmetry c ′ in C ′ then q = q ′− for N = 2 and q = q ′− for N >
2. In the first case we conclude as before sinceby Prop. 18.5 q and q − gives rise to equivalent tensor categories again. In the second casewe use the twist equation c ( X q , X q ) = θ X q ⊗ θ X q ◦ θ − X q ⊗ X q and similarly for X q ′ , θ ′ and c ′ ,which implies c ( X q , X q ) and c ′ ( X q ′ , X q ′ ) have the same eigenvalues. This implies q C = 1and therefore N + 1 < ℓ ≤ (cid:3) Note that the positivity assumption in Theorem 18.1 is redundant for N odd by unique-ness of the ribbon structure of every braided symmetry of C q,N,ℓ , Prop. 18.11. The followingexample shows that this assumption can not be dropped for N even. Example 18.14.
Consider C ( s l , q, ℓ ) for q = e iπ/ℓ with ℓ > ε +1 and ε + − described in Prop. 18.6. By Remark 18.8, the identity isomorphism betweenthe corresponding representation rings can not be induced by a braided tensor equivalence. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 103
On the other hand, each of the two braided categories has its own positive ribbon structure,say θ and θ − respectively. For an irreducible λ = a Λ we have θ ( λ ) = q a ( a +2) , θ − ( λ ) =( − q / ) a ( a +2) where q / = e iπ/ ℓ . If η ∈ (1 ,
1) is the natural monoidal transformation ofthe identity functor taking value − X then it follows from theproof of Prop. 18.11 that ηθ − = θ . But ηθ − is another ribbon structure for ε + − .We conclude the section with a partial result concerning ribbon equivalence of examplesof s l N,ℓ -type categories where pseudo-unitarity is not assumed but the ribbon structure isfixed.
Proposition 18.15.
Let q and q ′ ∈ C be either not non-trivial roots of unity or elsesquare to primitive roots of unity of order ℓ > N + 1 and let us endow both C ( s l N , q, ℓ ) and C ( s l N , q ′ , ℓ ) with some braided symmetry. If there is an isomorphism of based rings f : Gr ( C ( s l N , q, ℓ )) → Gr ( C ( s l N , q ′ , ℓ )) identifying the generating representations and com-patible with the canonical ribbon structures then q = q ′ and furthermore there is a ribbontensor equivalence F : C ( s l N , q, ℓ ) → C ( s l N , q ′ , ℓ ) inducing f .Proof. We write the respective braided symmetries c and c ′ as in the proof of the previoustheorem, where now q and q ′ are general. We again have that the canonical ribbon structureof C ( s l N , q, ℓ ) takes the form θ λ = q ±h λ,λ +2 ρ i . We need to be a bit more explicit on theexponents, so we write λ = P N − n j Λ j , where Λ j are the fundamental weights, n j arenon-negative integers and ρ = P N − Λ j . Then h Λ k , Λ j i = d j d k,j , where d k,j are such thatΛ k = P j d k,j α j , with α j the simple roots, d j = h α j ,α j i , hence equal to 1 in our case. Thisgives h λ, λ + 2 ρ i = P k,j n k ( n j + 2) d k,j . The matrix ( d k,j ) is given in Table 1 at pag. 69 of[68]. In particular one obtains h Λ k , Λ k + 2 ρ i = kN ( N − k )( N + 1), see e.g. Sect. 6 in [105],and more generally h n Λ k , n Λ k + 2 ρ i = n [ h Λ k , Λ k + 2 ρ i + ( n − d k,k ] = nN [ k ( N − i )( N + 1) + ( n − k ( N − k )] = nkN ( N − k )( N + n ) . Assuming again that θ corresponds to the plus sign, we have θ Λ = q N − N , θ = q N +2 − N and for N > θ Λ = q N − − N . It follows that θ θ − = q − N , θ θ − = q . We claim that we may assume that θ ′ corresponds to a plus sign as well, and we show itin the same way. The first equation gives q = q ′ for N = 2. Assuming N >
2, the secondequation gives q ′ = ± q or q ′ = ± iq . a) Case q ′ = − q . If N is odd then q and − q giverise to equivalent tensor categories by Prop. 18.5. We may thus assume with no loss ofgenerality that q = q ′ . If N is even then N − θ N Λ = θ ′ N Λ we may exclude q ′ = − q . b) We next show that the cases q ′ = ± iq are not realized. We need to compute
04 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI the ribbon structure of weights which are sums of different fundamental weights, and forthis we use the following addition formula which follows from bilinearity and symmetry ofthe inner product h Λ + λ, Λ + λ + 2 ρ i = h Λ , Λ + 2 ρ i + h λ, λ + 2 ρ i + 2 h λ, Λ i . On the other hand, the equation θ N Λ = θ ′ N Λ requires N odd, we may thus consider theweights µ = Λ N − and ν = Λ N +12 and since h Λ k , Λ i = kN we have h µ + ν, Λ i = 1 . Applying the addition formula to µ and ν in place of λ and comparing the ribbon structureon the weights in { Λ , µ, ν, Λ + µ, Λ + ν } leads to q h µ, Λ i = ( q ′ ) h µ, Λ i and q h ν, Λ i =( q ′ ) h ν, Λ i , hence after term by term multiplication we get ( q ′ ) = q , contradicting q = 0.Hence in all cases we may arrange q ′ = q . The relation θ Λ = θ ′ Λ now implies that alsothe corresponding two N -roots q /N and q ′ /N are the same, and we thus have a ribbontensor equivalence F : C → C ′ inducing f . (cid:3) Turning C ∗ -categories into tensor C ∗ -categories, II Let A be a discrete weak quasi bialgebra with a pre- C ∗ -algebra structure and let Ω ∈ M ( A ⊗ A ) be a given partially invertible operator with domain ∆( I ). We develop a criterionthat will be useful in Sect. 21, 23, 24 to verify the axioms of a positive Ω-involution.Let ρ ∈ Rep h ( A ) be a ∗ -representation. Since the coproduct is not coassociative ingeneral, there are different tensor powers of ρ each given order n ≥
3, but they are allequivalent.
Definition 19.1.
A representation ρ is called generating if ρ n ( a ) = 0 for all n implies a = 0, where ρ n denotes the choice of an n -th tensor power of ρ .It suffices to check the generating condition on a choice of a n -th tensor power of ρ foreach n .Let σ and τ be f.d. ∗ -preserving representations of A on Hlbert spaces. As for the case ofΩ-involutive weak-quasi bialgebras, we may define the sesquilinear form induced by Ω onthe tensor product space and consider the ρ ⊗ σ as a representation on this space, except wedo not know whether it is a Hilbert space ∗ -representation. Let ρ be a generating Hilbertspace ∗ -representation. We may consider the full subcategory C ρ of Rep( A ) with objectsthe various tensor powers ρ n of ρ on sesquilinear spaces. This is a tensor category. Wemay determine the Hermitian form of ρ n with an inductive procedure, as follows. Let Ω n be the element of A ⊗ n defining this form via ( ξ, η ) = ( ξ, Ω n η ) p , where ( ξ, η ) p denotes theuntwisted n -th tensor power of the original Hermitian form of ρ on V ⊗ nρ . Let ∆ n : A → A ⊗ n denote the homomorphism defining the A -action on the space of ρ n . Writing ρ n = ρ r ⊗ ρ s , with r + s = n, r, s < n, we have that Ω n = Ω r ⊗ Ω s ∆ r ⊗ ∆ s (Ω) , ∆ n = ∆ r ⊗ ∆ s ◦ ∆ , EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 105 where Ω = I , Ω = Ω, ∆ = 1, ∆ = ∆.Assume for a moment that Ω is an Ω-involution. Then we inductively get the followingrelations, extending (8.1)–(8.3).Ω n = Ω ∗ n , (19.1)Ω − n Ω n = ∆ n ( I ) , Ω n Ω − n = ∆ n ( I ) ∗ , (19.2)∆ n ( a ) ∗ Ω n = Ω n ∆ n ( a ∗ ) , a ∈ A. (19.3)We next go back to the original situation, then we only know that the above relationholds under the image of ρ n if we already know that ρ n is a ∗ -representation. Theorem 19.2.
Let A be a discrete pre-C*-algebra equipped with the structure of a weakquasi-bialgebra, and let ρ be a generating C*-representation of A . Let Ω ∈ M ( A ⊗ A ) bea partially invertible element with domain ∆( I ) and such that for every irreducible C ∗ -representation σ , σ ⊗ ρ (Ω) , σ ⊗ ρ (Ω − ) , ρ ⊗ σ (Ω) , ρ ⊗ σ (Ω − ) (19.4) are positive on the full tensor product space, that σ ⊗ ρ and ρ ⊗ σ are C ∗ -representationsw.r.t. the Ω -twisted inner product and that σ ⊗ ρ ⊗ ρ ( I ⊗ Ω1 ⊗ ∆(Ω)) , and ρ ⊗ ρ ⊗ σ (Ω ⊗ I ∆ ⊗ are positive as well. Moreover, assume that the associativity morphisms σ ⊗ ρ ⊗ ρ (Φ) , ρ ⊗ ρ ⊗ σ (Φ) are unitary with respect to the Ω -twisted inner products. Then Ω is a positive element of M ( A ⊗ A ) and in this way A becomes a unitary discrete weak quasi bialgebra and Ω isuniquely determined by the operators σ ⊗ ρ (Ω) for every irreducible σ .Proof. It follows from the first relation in (19.4) that σ ⊗ ρ ⊗ ρ (Ω ⊗ I ∆ ⊗ ⊗ I ∆ ⊗ I ⊗ Ω1 ⊗ ∆(Ω) are positive on V ρ ⊗ V σ ⊗ V ρ and V ρ ⊗ V ρ ⊗ V σ . Every associativity morphism α ρ r ,ρ s ,ρ t = ρ r ⊗ ρ s ⊗ ρ t (Φ) of the fullsubcategory C ρ of Rep( A ) with objects parenthisized tensor powers of ρ can be written asa composition of tensor products with identity of morphisms of the form α ρ r ,ρ,ρ , α ρ,ρ r ,ρ , α ρ,ρ,ρ r . By complete reducibility of representations and naturality, our assumptions implyunitarity of the first and the last, and the pentagon equation implies unitarity of the middleone. It follows that the associators imply that α ρ r ρ s ρ t are unitary. We next show that every ρ n is a C*-representation for the choice iteratively defined by ρ n +1 = ρ n ⊗ ρ . Assuming thata fixed ρ n is so, we decompose ρ n into pairwise orthogonal irreducible components σ . Since V σ ⊗ V ρ is invariant under σ ⊗ ρ (Ω), ρ n ⊗ ρ (Ω) is positive on V ρ n ⊗ V as well, hence it is apositive element of the C*-algebra ρ n ( A ) ⊗ ρ ( A ). We may thus find an element S ∈ A ⊗ A such that ρ n ⊗ ρ ( S ) is selfadjoint and ρ n ⊗ ρ (Ω) = ρ n ⊗ ρ ( S ) . On the other hand, theHermitian form of ρ n +1 is defined by the action of the operator ρ ⊗ n +1 [Ω n +1 ] on V ⊗ n +1 ρ withΩ n +1 = Ω n ⊗ I ∆ n ⊗ ρ ⊗ n +1 [Ω n +1 ] = ρ ⊗ n [Ω n ] ⊗ Iρ n ⊗ ρ (Ω) =
06 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI ρ ⊗ n [Ω n ] ⊗ Iρ n ⊗ ρ ( S ) = ρ ⊗ n +1 [Ω n ⊗ I ∆ n ⊗ S ) ] = ρ ⊗ n +1 [∆ n ⊗ S ) ∗ Ω n ⊗ I ∆ n ⊗ S )] = ρ ⊗ n +1 [∆ n ⊗ S )] ∗ ρ ⊗ n [Ω n ] ⊗ Iρ ⊗ n +1 [∆ n ⊗ S )]and this is a positive operator by positivity of ρ ⊗ n [Ω n ]. We consider the C*-representation τ = ⊕ n ρ n , which is faithful as ρ is generating. We are left to show that τ ⊗ τ [Ω] is a positiveoperator in this representation, since it will then be a positive element of τ ( A ) ⊗ τ ( A ), andtherefore Ω positive in A ⊗ A . To this aim, we observe that the action of τ ⊗ τ [Ω] on thesubspace V ρ r ⊗ V ρ s is given by that of ρ ⊗ n (Ω ′ n ), where n = r + s and Ω ′ n = Ω r ⊗ Ω s ∆ r ⊗ ∆ s (Ω).Thanks to unitarity of the associativity morphisms and an inductive argument we see that ρ ⊗ n (Ω ′ n ) = ρ ⊗ n (Φ n Ω n Φ ∗ n ) for suitable associativity morphisms Φ n . It follows that τ ⊗ τ (Ω)is positive, hence Ω is positive in M ( A ⊗ A ). Therefore C ρ is a unitary tensor categorywith unitary structure defined by Ω. Now the axioms of the Ω-involution on A follow. (cid:3) The above theorem will be useful in the construction of the main examples of Sect. 24.
Remark 19.3.
For example, if A is a finite dimensional C*-algebra A = L r M n r ( C )and ρ is generating, every ρ r ⊗ ρ is unitarily equivalent to an orthogonal direct sum of theprojection C*-representations ρ s : A → M n s ( C ) and their opposites ρ − s by Prop. 9.9. Bythe previous theorem, verification of positivity of Ω reduces to the question of whether thenegative forms ρ − s can be ruled out for this subclass of fusion tensor products.We conclude the section with a further discussion on C ∗ -transportability. In comparisonwith Sect. 12, the following discussion gives a direct method to transport the tensorstructure from C to C + that will be useful in Sect. 21, 23, 24. We note however thatthis method is already implicit in our main results Theorem 12.6 and 12.7. Let us assumecodition a). It is not difficult to see, using a quasi-inverse of F , that when C has a weakdimension function there always is a faithful weak quasi-tensor functor G : C → Hilb suchthat GF is a ∗ -functor. Recall from Remark 9.10 that there are examples for which thetensor structure of C is not transportable to C + and in these cases we have a functor G which does not take the associativity morphisms to unitary morphisms. On the other hand,it follows from Theorem 12.6 that when the tensor structure of C is C*-transportable to C + then we may find G taking the associativity morphisms to unitary morphisms. Thefollowing proposition shows that the converse holds. Proposition 19.4.
Let F : C + → C satisfy a) and assume that C admits a weak dimensionfunction. Let G : C → Hilb be a faithful functor such that G + = GF is a ∗ -functor and themorphisms G ( α ρ,σ,τ ) are unitary. Then every weak quasi-tensor structure on G induces thestructure of a tensor C ∗ -category on C + s.t. F is a tensor equivalence (C*-transportability).Proof. Let (
F, G ) be a weak quasi-tensor structure for G , thus F G = 1 and also G ∗ F ∗ = 1.The functors G , G + correspond to the forgetful functors of a compatible triple as in Def.12.2. We consider the corresponding weak quasi bialgebra ( A, ∆ , Φ) with A = Nat ( G ).The linear equivalence F : C + → C induces an algebra isomorphism A → A + = Nat ( G + ), η → η F ( ) . Since G + is a ∗ -functor, A + is a C ∗ -algebra. By Theorem 12.6 we only need EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 107 to make A + into a unitary weak quasi-bialgebra. We introduce the structure similarly tothe case of the Tannakian theorems 5.6, 10.5. We denote by x , y , z, . . . the irreduciblerepresentations of A + and define˜ F x.y := F F ( x ) , F ( y ) : G ( F ( x )) ⊗ G ( F ( y )) → G ( F ( x ) ⊗ F ( y ))˜ G x.y := G F ( x ) , F ( y ) : G ( F ( x ) ⊗ F ( y )) → G ( F ( x )) ⊗ G ( F ( y )) . This suffices to make A + into a weak quasi-bialgebra by˜∆( η F ( ) ) x,y = ˜ G x,y η F ( x ) ⊗ F ( y ) ˜ F x,y , ˜Φ x,y,z = Φ F ( x ) , F ( y ) , F ( z ) =1 ⊗ ˜ G y,z ◦ G F ( x ) , F ( y ) ⊗ F ( z ) ◦ G ( α F ( x ) , F ( y ) , F ( z ) ) ◦ F F ( x ) ⊗ F ( y ) , F ( z ) ◦ ˜ F x,y ⊗ . We introduce an Ω-involution on A + by Ω = ˜ F ∗ ˜ F , Ω − = ˜ G ˜ G ∗ . In this more generalsetting the only non-trivial verification is axiom (8.5) which reduces to G ( α F ( x ) , F ( y ) , F ( z ) ) ∗ = G ( α − F ( x ) , F ( y ) , F ( z ) )and holds by assumption. (cid:3) Coboundary categories and Deligne’s theorem
By an interesting result of Deligne, the study of dimension in a braided tensor categorycan be addressed in two equivalent ways: via right duality with extra (pivotal/spherical)structure or else via extra structure on the braided symmetry (balancing/ribbon structure).In this section we introduce a notion of symmetry that is more general than that ofbraided symmetry, and we call generalised coboundary . It is a generalisation of both thenotion of braided symmetry and that of a coboundary due to Drinfeld that allows to studythese symmetries in a unified way.The generalisation is motivated by the fact that some of the structures that we study inthis paper do not need the full notion of a braided symmetry, but only the more generalclass of symmetries, which have the advantage of stability under certain twist deformation.A important source of coboundaries indeed arises from deformation of braided symme-tries with ribbon structure and plays a central role in the unitary structure of the weakquasi-Hopf algebras arising from quantum groups at roots of unity studied in Sects. 21, 24.We study pivotal or spherical structures in tensor categories with a generalised cobound-ary, and we extend Deligne result to this case. We start reviewing the notion of pivotaland spherical category.If ρ ∨ is a two-sided dual of ρ and if ( b ρ , d ρ ) and ( b ′ ρ , d ′ ρ ) respectively solve the rightand the left duality equations for this pair, then we can associate two functionals on themorphism space ( ρ, ρ ), called left and right quantum traces, viaTr Lρ ( T ) = d ρ ◦ ρ ∨ ⊗ T ◦ b ′ ρ (20.1)Tr Rρ ( T ) = d ′ ρ ◦ T ⊗ ρ ∨ ◦ b ρ . (20.2)
08 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI
If these solutions correspond to pivotal (or spherical) structures a well behaved theory ofdimension can be developed. We briefly recall the main aspects, dropping, for simplicity,the associativity morphisms in most of our formulae in this section.Let ( ρ ∨ , b ρ , d ρ ) be a right duality, see Sect. 3, and D : C → C the associated functor asin (3.6). Note that D : C → C is a covariant tensor functor. We assume from now on thatour category has two-sided duals, so there is a natural isomorphism u from the identityfunctor 1, to D , which, however, need not be monoidal. An example of this occurrencearises in the framework of representations of semisimple weak quasi-Hopf algebras. Thecategory has two-sided duals if the square of the antipode S is an inner automorphism. Thenatural isomorphism is monoidal if the implementing element can be chosen group-like,but this is not always the case. On the other hand, any natural isomorphism u ∈ (1 , D )in a category with two-sided duals defines a left duality structure coinciding with the rightone on the objects via b ′ ρ = 1 ρ ∨ ⊗ u − ρ ◦ b ρ ∨ , d ′ ρ = d ρ ∨ ◦ u ρ ⊗ ρ ∨ . (20.3)Furthermore any pair of right and left dualities ( ρ ∨ , b ρ , d ρ ) and ( ∨ ρ, b ′ ρ , d ′ ρ ) with ρ ∨ = ∨ ρ isof this form with u uniquely determined. Indeed, the morphism u ρ := d ′ ρ ⊗ ρ ∨∨ ◦ ρ ⊗ b ρ ∨ (20.4)is a natural isomorphism in (1 , D ) with u − ρ = d ρ ∨ ⊗ ρ ◦ ρ ∨∨ ⊗ b ′ ρ and the two constructionsare inverse of one another. Given u ∈ (1 , D ), any other ω ∈ (1 , D ) can be written in theform ω = uv − , with v ∈ (1 ,
1) uniquely determined. (The use of the inverse of v matchesour notation in the framework of quantum groups, and originates with the convention in[122]). Correspondingly, any other left duality is of the form˜ b ρ = 1 ρ ∨ ⊗ v ρ ◦ b ′ ρ , ˜ d ρ = d ′ ρ ◦ v − ρ ⊗ ρ ∨ . (20.5)A pivotal structure on C is the datum of a right duality functor D together with a monoidal isomorphism ω ∈ (1 , D ) [45]. In a tensor category with right duality ( b ρ , d ρ ) the left dualitydefined by a pivotal structure ω in place of u in (20.3) will be denoted as ( b ℓρ , d ℓρ ). A pairof dualities ( b ρ , d ρ ) and ( b ℓρ , d ℓρ ) so related induces C -valued left and right quantum traces(20.1), (20.2) which are multiplicative on tensor product morphisms. A spherical structure on C is a pivotal structure such that the associated left and right traces coincide. In thiscase we shall simply write Tr ρ . A spherical category is a tensor category endowed with aspherical structure. In a spherical category Tr ρ ( ST ) = Tr σ ( T S ), for any pair of morphisms T ∈ ( ρ, σ ), S ∈ ( σ, ρ ) the categorical (or quantum) dimension ρ → d ( ρ ) is defined by d ( ρ ) = Tr ρ (1 ρ ) . It is additive, multiplicative and, for categories over C as those of this paper, it takes realvalues on the objects, see [8] and Sect. 2 in [38] for more details. It is not known whethera fusion category always admits a pivotal structure, but see [38, 97, 92] for results andreferences. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 109
Definition 20.1. A generalised coboundary is a natural isomorphism c ( ρ, σ ) ∈ ( ρ ⊗ σ, σ ⊗ ρ )satisfying (3.7) and such that the following diagram commutes.( ρ ⊗ σ ) ⊗ τ α −−−→ ρ ⊗ ( σ ⊗ τ ) ⊗ c −−−→ ρ ⊗ ( τ ⊗ σ ) c ⊗ y y c ( σ ⊗ ρ ) ⊗ τ c −−−→ τ ⊗ ( σ ⊗ ρ ) α ←−−− ( τ ⊗ σ ) ⊗ ρ (20.6)If c ( ρ, σ ) is a generalised coboundary then c ′ ( ρ, σ ) := c ( σ, ρ ) − is too. Example 20.2.
A generalised coboundary for which c satisfies the symmetry condition c = 1 is a coboundary in the sense introduced by Drinfeld [34]. Remark 20.3.
Every braided symmetry is a generalised coboundary. Indeed, if c is such asymmetry, we may use the hexagonal equations (3.8), (3.9) and verification of commutativ-ity of (20.6) reduces to the Yang-Baxter relation, which follows from the braided symmetryaxioms, see e.g. Prop. 8.1.10 in [37].The following statement explains the notion of generalised coboundary in an importantclass of tensor categories. Proposition 20.4.
Let A be a weak quasi bialgebra and Q ∈ A ⊗ A a twist such that A Q = A op . Then c ( ρ, σ ) := Σ ρ ⊗ σ ( Q ) is a generalised coboundary of Rep( A ) . We refer to (7.2), (7.3), (7.6), (7.15), with Q in place of R , for an explicit form of theequality A Q = A op . Remark 20.5.
The construction of generalised cobounderies on Rep( A ) of 20.4 extendsto the case where A is a discrete weak quasi bialgebra, and the twist between A and A op satisfies Q ∈ M ( A ⊗ A ). In this case, we also see that all generalised coboundaries ofRep( A ) are of this form, via Tannaka-Krein duality.We introduce twist deformation of generalised coboundaries. Proposition 20.6.
Let c be a generalised coboundary and η ∈ (1 , a natural isomorphismof the identity functor with η ι = 1 ι . Then c η ( ρ, σ ) := c ( ρ, σ ) ◦ η − ρ ⊗ η − σ ◦ η ρ ⊗ σ is a generalisedcoboundary as well. If c is a braided symmetry, c η may fail to be a braided symmetry, but it is a generalisedcoboundary. Proposition 20.7.
Let C be a tensor category with generalised coboundary c . Then dualsare two-sided. If the category has right duals and ( ρ ∨ , b ρ , d ρ ) denotes a right duality then b ′ ρ = c ( ρ ∨ , ρ ) − ◦ b ρ , d ′ ρ = d ρ ◦ c ( ρ, ρ ∨ ) (20.7) is a left duality with ρ ∨ = ∨ ρ . Conversely, given a left duality, ( ∨ ρ, b ′ ρ , d ′ ρ ) the same formuladefines a right duality.Proof. The left duality relations for b ′ ρ and d ′ ρ follow from a computation that uses, inorder, commutativity of the diagram (20.6), naturality of c and the right duality equationsfor b ρ and d ρ . (cid:3)
10 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI
Remark 20.8.
Note that we may apply the same construction to c ′ and get another leftduality with ∨ ρ = ρ ∨ , b ′′ ρ = c ( ρ, ρ ∨ ) ◦ b ρ , d ′′ ρ = d ρ ◦ c ( ρ ∨ , ρ ) − . (20.8)In the special case where c is a genuine coboundary, these two left dualities coincide, thusevery right duality ( b ρ , d ρ ) has a canonically associated left duality in this way. It is alsoeasy to see that the associated right and left traces coincide thanks to naturality of c .We do not know whether this pair of dualities corresponds to a pivotal structure for allcoboundaries, but this is known to be the case when c is a permutation symmetry or forall the examples of coboundaries that may be constructed from braided symmetries andtwist deformation. Definition 20.9.
Let C be a tensor category and let a ρ,σ ∈ ( ρ ⊗ σ, ρ ⊗ σ ) be a tensorstructure for the identity functor.a) A balancing structure for a is a natural isomorphism v ∈ (1 ,
1) making the identityfunctor 1 with tensor structure a monoidally isomorphic to the trivial tensor structure of1, so a ρ,σ = v ρ ⊗ v σ ◦ v − ρ ⊗ σ . (20.9)b) If C has a right duality ( ρ ∨ , b ρ , d ρ ), a ribbon structure for a is a balancing structurecompatible with duality, see Def. 3.3. Remark 20.10. If v is a balancing structure for a , the relation v ι = 1 ι easily follows fromthe fact that we are assuming that ι is a strict unit, but for general categories it needs tobe part of the axioms.We next see that the question of whether a rigid tensor category with a generalisedcoboundary admits a pivotal or spherical structure can be reduced to the analysis oftwo tensorial structures of the identity functor, which are naturally associated to thecoboundary. In the case where c is a braided symmetry, these reduce to the same structure,but they may be distinct in general. We first generate tensor structures of 1 from c . Definition 20.11.
Let c be a generalised coboundary and ( b ρ , d ρ ) a right duality. Considerthe left duality ( b ′ ρ , d ′ ρ ) described in (20.7). The natural isomorphism u ∈ (1 , D ) associatedto this pair as in (20.3), (20.4) is called Drinfeld isomorphism .Hence Drinfeld isomorphism is the composite u ρ : ρ ⊗ b ρ ∨ −−−→ ρ ⊗ ρ ∨ ⊗ ρ ∨∨ c ⊗ −−→ ρ ∨ ⊗ ρ ⊗ ρ ∨∨ d ρ ⊗ −−−→ ρ ∨∨ . (20.10) Proposition 20.12.
Let c be generalised coboundary. The isomorphisms c ( ρ, σ ) := c ( σ, ρ ) ◦ c ( ρ, σ ) ∈ ( ρ ⊗ σ, ρ ⊗ σ ) define a tensor structure on the identity functor.Proof. Naturality of c in the two variables is obvious. The tensor structure axiom c ( ρ, στ ) ◦ ρ ⊗ c ( σ, τ ) = c ( ρσ, τ ) ◦ c ( ρ, σ ) ⊗ τ is indeed a simple consequence of the generalised coboundary axioms for c . (cid:3) EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 111
We start with a condition leading to the construction of two coinciding quantum traces.
Theorem 20.13.
Let c be a generalised coboundary, ( b ρ , d ρ ) a right duality and v ∈ (1 , a ribbon structure for c . Then the left and right quantum traces corresponding to a givenright duality ( b ρ , d ρ ) and to the associated left duality via ω := uv − ∈ (1 , D ) , coincide.Proof. The left duality defined by ω = uv − is given by˜ b ρ = 1 ρ ∨ ⊗ v ρ ◦ c ( ρ ∨ , ρ ) − ◦ b ρ , ˜ d ρ = d ρ ◦ c ( ρ, ρ ∨ ) ◦ v − ρ ⊗ ρ ∨ . (20.11)The corresponding right trace is given byTr Rρ ( T ) = d ρ ◦ ρ ∨ ⊗ T ◦ c ( ρ, ρ ∨ ) ◦ v − ρ ⊗ ρ ∨ ◦ b ρ . To compare it with the left trace we computeTr Lρ ( T ) = d ρ ◦ ρ ∨ ⊗ T ◦ ρ ∨ ⊗ v ρ ◦ c ( ρ ∨ , ρ ) − ◦ b ρ = d ρ ◦ ρ ∨ ⊗ T ◦ c ( ρ ∨ , ρ ) − ◦ v ρ ⊗ ρ ∨ ◦ b ρ = d ρ ◦ ρ ∨ ⊗ T ◦ c ( ρ ∨ , ρ ) − ◦ ρ ⊗ v ρ ∨ ◦ b ρ , the last equality follows from v ρ ⊗ ρ ∨ ◦ b ρ = 1 ρ ⊗ v ρ ∨ ◦ b ρ in turn due to compatibility of v with duality. On the other hand, c ( ρ ∨ , ρ ) − ◦ ρ ⊗ v ρ ∨ = c ( ρ, ρ ∨ ) ◦ v − ρ ⊗ ρ ∨ ◦ v ρ ⊗ ρ ∨ thanksto the balancing condition c ( ρ, σ ) = v ρ ⊗ v σ ◦ v − ρ ⊗ σ . The conclusion now follows from thisand naturality of v . (cid:3) We have yet another tensor structure of the identity functor induced by c as follows.Let d ( ρ, σ ) : ρ ∨∨ ⊗ σ ∨∨ → ( ρ ⊗ σ ) ∨∨ denote the natural tensor structure of D . In theframework of weak quasi-Hopf algebras, we have explicitly computed the element of A ⊗ A inducing d , see Sect. 5. We can equip 1 with the new tensor structure, denoted c ,obtained pulling back the tensorial structure of D via Drinfeld isomorphism. In otherwords, we let c ( ρ, σ ) ∈ ( ρ ⊗ σ, ρ ⊗ σ ) denote the isomorphisms making the followingdiagram commute, ρ ⊗ σ u ρ ⊗ u σ −−−−→ ρ ∨∨ ⊗ σ ∨∨ c y y d ρ ⊗ σ u ρ ⊗ σ −−−→ ( ρ ⊗ σ ) ∨∨ (20.12)We next analyse dependence of Drinfeld isomorphism and c on the right duality. Lemma 20.14.
Let ( ρ ∨ , b ρ , d ρ ) and ( ˜ ρ, ˜ b ρ , ˜ d ρ ) be two right dualities with associated functors D and ˜ D respectively, and let ξ ∈ ( ˜ D, D ) the corresponding monoidal isomorphism. Let u and ˜ u be corresponding Drinfeld isomorphisms defined by the same generalised coboundary.Then ˜ u ρ = ζ ρ ◦ u ρ where ζ ρ := ξ − ρ ◦ ξ ∨ ρ : D → ˜ D is the composite monoidal isomorphism.Proof. The proof follows from a computation starting from ˜ u ρ taking into account b ˜ ρ = ξ − ρ ⊗ ξ ∨ ρ ◦ b ρ ∨ , ˜ b ˜ ρ = 1 ˜ ρ ⊗ ξ − ρ ◦ b ˜ ρ , ˜ d ρ = d ρ ◦ ξ ρ ⊗ ρ and naturality of c . (cid:3)
12 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI
Proposition 20.15.
Let C be a tensor category with generalised coboundary c and rightduality ( ρ ∨ , b ρ , d ρ ) . Then the isomorphism c ( ρ, σ ) ∈ ( ρ ⊗ σ, ρ ⊗ σ ) is a tensor structure of the identity functor which does not depend on the choice of theright duality. Remark 20.16. a) It is known that c = c if c ( ρ, σ ) is a braided symmetry in a stricttensor category, for a proof see [37], Prop. 8.9.3. b) In Prop. 7.9 we have shown that c = c for the braided symmetry associated to the quasitriangular structure of any w-Hopf algebra.We give an example showing that c and c may be different tensor structures. Example 20.17.
Consider the tensor category C = Vec G of finite dimensional G -gradedvector spaces over a finite abelian group G , with tensor product defined in the standardway, for V = ( V g ) and W = ( W h ), ( V ⊗ W ) k = ⊕ gh = k V g ⊗ V h , and natural associator, see[37]. Then every group element g defines a 1-dimensional space δ g of grade g and these areall the irreducible objects up to equivalence. We have that δ − g is both a right and left dualof δ g and duality equations are solved by the identity maps. A generalised coboundary c is determined by the action on δ g ⊗ δ h , and this gives a complex-valued nonzero function c ( g, h ) on two variables. The coboundary relation corresponds to requiring that c ( g, h )be a two-cocycle: c ( g, h ) c ( gh, k ) = c ( h, k ) c ( g, hk ) with c (1 , g ) = c ( g,
1) = 1. Drinfeldisomorphism u g acts as c ( g, g − ) on δ g , while d acts trivially. It follows that c ( g, h ) = c ( g, g − ) c ( h, h − ) c ( gh, ( gh ) − ) − while c ( g, h ) = c ( g, h ) c ( h, g ). A computation shows that c = c if and only if c ( h, h − ) = c ( h, g ) c ( h, ( gh ) − ), and it is easy to see that this is notalways the case for a normalised cohomologically trivial c ( g, h ) = µ ( gh ) µ ( g ) − µ ( h ) − for G = Z .The following extends Deligne’s result to generalised coboundaries. Theorem 20.18.
Let C be a tensor category with generalised coboundary c and right duality ( ρ ∨ , b ρ , d ρ ) . There is a bijective correspondence between pivotal structures ω ∈ (1 , D ) andbalancing structures z ∈ (1 , for c given by ω = uz − , where u ∈ (1 , D ) is Drinfeld isomorphism associated to c .Proof. The map z → ω = uz − is a bijective correspondence between isomorphisms ω ∈ (1 , D ) and z ∈ (1 , ω is monoidal precisely when z is a balancing for c , bycommutativity of (20.12). (cid:3) We derive a sufficient condition for existence of spherical structures.
Corollary 20.19.
Let C be a tensor category with right duality ( b ρ , d ρ ) , generalised cobound-ary c satisfying c ( ρ, σ ) = c ( ρ, σ ) . (20.13) EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 113 and ribbon structure v . Then the pivotal structure ω = uv − is spherical. The correspondingleft duality is given by b ℓρ = 1 ρ ∨ ⊗ v ρ ◦ c ( ρ ∨ , ρ ) − ◦ b ρ , d ℓρ = d ρ ◦ c ( ρ, ρ ∨ ) ◦ v − ρ ⊗ ρ ∨ (20.14) Proof.
This is a consequence of Prop. 20.19 and Prop. 20.18. The left duality equationsfollow from (20.5), (20.7). (cid:3)
In particular, the quantum dimension is given by d ( ρ ) = d ρ ◦ ρ ∨ ⊗ v ρ ◦ c ( ρ ∨ , ρ ) − ◦ b ρ = d ρ ◦ c ( ρ, ρ ∨ ) ◦ v − ρ ⊗ ρ ∨ ◦ b ρ . (20.15)Prop. 20.19 recovers corresponding results known for ribbon categories [122]. Note thatCor. 20.19 is of little use in the case where (20.13) does not hold. Indeed in the Example20.17 as we may choose for c the unique permutation symmetry, so c = c = 1, gives thatthe associated Drinfeld isomorphism u ρ = 1 is a spherical structure.We next discuss properties of twisted generalised coboundaries. Let ( b ρ , d ρ ) be a fixedright duality, c a generalised coboundary, u the associated Drinfeld isomorphism and c ∈ ( ρ ⊗ σ, ρ ⊗ σ ) natural isomorphism as in (20.12). Let η ∈ (1 ,
1) be a natural isomor-phism, and c η the twisted coboundary. The corresponding isomorphisms will be denotedrespectively by u η and c η . Proposition 20.20.
Let η ∈ (1 , be a compatible with duality. We have a) u η = u ◦ η − , b) c η ( ρ, σ ) = η ρ ⊗ σ ◦ c ( ρ, σ ) ◦ η − ρ ⊗ η − σ , (( c η ) ( ρ, σ ) = η ρ ⊗ σ ◦ c ( ρ, σ ) ◦ η − ρ ⊗ η − σ , )c) if v is a balancing (ribbon) structure for c ( c ) then v η := v ◦ η − is a balancing(ribbon) structure for c η (( c η ) )d) v and v η correspond to the same pivotal structure under the map described in Prop.20.18, and therefore to the same left duality and quantum traces, e) if c satisfies (20.13) then so does c η .Proof. a) The proof follows from a computation starting from (20.10), with c replaced by c η , where we use naturality and compatibility with duality of η and the fact that the rightduality functor (3.6) can equivalently be defined by d σ ◦ σ ∨ ⊗ T = d ρ ◦ T ∨ ⊗ ρ . Theremaining statements follow from one another. (cid:3) We describe a twisting making a generalised coboundary with a balancing structure intoa genuine coboundary and Drinfeld isomorphism into a monoidal isomorphism from theidentity tensor functor. This twisting first appeared in the work of Drinfeld [34] in theframework of quantised universal Hopf algebras. As it turns out, the associated sphericalstructure is the same as that arising in the framework of ribbon categories.
Theorem 20.21.
In a tensor category with right duality, let c be a generalised coboundarysatisfying (20.13) (e.g. a braided symmetry) with balancing structure v , and let w ∈ (1 , be a natural isomorphism compatible with duality such that w = v . Then c w is a cobound-ary, c w = 1 ⊗ , v w = 1 , and u w is a spherical structure coinciding with that defined by c and v as in Prop. 20.18.
14 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI
The construction of c w is the analogue of Drinfeld construction of unitarized R -matrixin a ribbon Hopf algebra. In Sect. 21 we shall study the relation with Ω-involution.21. Hermitian coboundary wqh algebras
In this section we introduce the notion of
Hermitian coboundary weak quasi Hopf algebra .Essentially, we understand these as as ribbon weak quasi-Hopf algebras endowed with a ∗ -algebra structure satisfying various compatibility relations between the ∗ -involution, thecoproduct and ribbon structure. We are mainly interested in the case of discrete algebraswith a pre- C ∗ -algebra structure.The most relevant structural aspect of our definition is the relation between coproductand ∗ -involution. Informally, this relation may be interpreted as an antimultiplicativityproperty of the involution on the ‘dual noncommutative function algebra’, that is ( AB ) ∗ = B ∗ A ∗ . When we take the adjoint on both sides, we get an equation that dually identifies theopposite coproduct ∆ op and the adjoint coproduct ˜∆. To be more precise, we require that∆ op and ˜∆ (together with all the remaining structural data) are related by a trivial twist.Moreover, since we have an R -matrix which relates ∆ op and ∆, we may interpret thatnoncommutativity arises explicitly from the R -matrix as is familiar in quantum grouptheory. This property makes these algebras rather different from the ordinary Hopf ∗ -algebras, where coproduct and ∗ -involution commute.Among other axioms we assume a relation involving directly the unitary structure withthe braiding, or more precisely with the coboundary symmetry in the representation cate-gory. We assume the existence of a square root of the ribbon structure. Thus we have anassociated coboundary in the representation category in the sense of Sect. 20. It followsfrom the axioms that there is an Ω-involution on the algebra in the sense of Sect. 8 associ-ated to the braiding data. When the Ω-involution of an Hermitian coboundary weak quasiHopf algebra it is unitary, we shall talk of a unitary coboundary weak quasi-Hopf algebra .In this section we study the main properties. For example, among general Ω-involutions,those associated to a coboundary always make the braiding unitary, see Theorem 21.9.Moreover, we shall give a characterization of the case where an Hermitian coboundaryweak quasi-Hopf algebra gives rise to an Hermitian ribbon category, Theorem 21.13.The main result of this section is a Tannakian characterization of Hermitian cobound-ary weak quasi-Hopf algebras, see Theorem 22.1. This characterization describes suchalgebras as categories endowed with a faithful functor to Herm with a weak quasi-tensorstructure ( F, G ) and compatibility equations between the coboundary of the category, thepermutation symmetry of Herm and (
F, G ). The simplest case is that of symmetric tensorcategories, and the Tannakian characterization becomes the notion of symmetric tensorfunctor. In particular compact groups is a natural class of examples, and we are in thesetting of the Doplicher-Roberts duality theorem [31]. More generally, the permutationsymmetry is replaced by the coboundary of Drinfeld in the sense of Sect. 20.In the next section we discuss a possibly proper subclass of Hermitian coboundary weakquasi-Hopf algebras and we shall develop a criterion to construct such algebras.
EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 115
We shall show in the next section that the unitarization of a unitary coboundary weakquasi Hopf algebra in this subclass, is again an algebra of this kind with the advantagethat both the unitary structure and the R -matrix take a simpler form, see Remark 22.4completely determined by the square root of the ribbon structure. Somewhat remarkablyto us, it seems to remind the form taken by Drinfeld R -matrix of the quasi-Hopf algebraassociated to Knizhnik-Zamolodchikov differential equations in [34]. It seems valuable tous that this simple R -matrix may be derived in a general setting by the study of unitarystructures of ribbon weak quasi Hopf algebras. We hope to further develop this study infuture updates of this paperOur interest in discrete algebras is motivated by the unitary structure of the fusioncategories C ( g , q, ℓ ) associated to U q ( g ) at certain roots of unity. Kirillov defined a tensor ∗ -category tensor equivalent to C ( g , q, ℓ ) and conjectured that these where unitary. Theconjecture was shown to be true by Wenzl and Xu [79], [128], [134]. We may regard ournotion as an abstract version of Kirillov ∗ -structure following the approach of Wenzl in[128]. We shall recall these results in Sect. 24 and we recall in particular that the mainexample of Hermitian coboundary weak quasi-Hopf algebra is U q ( g ) itself for | q | = 1,although not a semisimple example at roots of unity.Furthermore in Sect. 24 we shall construct f.d. unitary coboundary w-Hopf algebras assuitable quotients of U q ( g ) with representation category equivalent to C ( g , q, ℓ ).Recall that for a general weak quasi-bialgebra A we have defined a twisted algebra A F , see Prop. 4.8, the opposite algebra A op , see (7.1) and furthermore, if A is also a ∗ -algebra, we have introduced the adjoint algebra ˜ A in (7.8). Note that A F , A op , and ˜ A have quasitriangular structures naturally induced by one of A , see Prop. 7.2. Moreover, A op and ˜ A have a strong antipode if so does A , and similarly for A F if (4.14) holds. Inparticular, A op and ˜ A are w-bialgebras if so is A , and similarly for A F if F is a 2-cocycle. Definition 21.1. A Hermitian coboundary weak quasi-Hopf algebra A is defined by thefollowing data:a) A weak quasi-Hopf algebra A endowed with a ∗ -algebra involution with an antipode( S, α, β )b) a ribbon structure (
R, v ) for A associated to ( S, α, β ) (see Def. 7.5) such that theribbon element v ∈ A is unitary,c) a unitary central square root w ∈ A of v such that ε ( w ) = 1, S ( w ) = w ,d) ˜ A = ( A op ) E as quasitriangular weak quasi-bialgebras, where E = ∆( I ) ∗ ∆ op ( I ) is atrivial twist, that is E − = ∆ op ( I )∆( I ) ∗ . Remark 21.2.
Our axioms are motivated by the structure of U q ( g ) for | q | = 1 that willbe important to us, and we shall recalled it in Sect. 24, Theorem 24.1. Notice howeverthat, since the R -matrix and ribbon structure lie in a suitable topological completion of U q ( g ) ⊗ U q ( g ) [114], this algebra can not be included as an example, unless we weaken ouraxioms. However we shall refrain from doing this. To deal with examples where the ribbonstructure lies in a larger algebra, we shall content to consider the case of discrete algebras.
16 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI
Definition 21.3. A discrete Hermitian coboundary weak quasi-Hopf algebra is defined bya discrete weak quasi-Hopf algebra A endowed with data ( ∗ , R, v, w ) such that axioms a)-d)hold as before for discrete algebras, that is the ∗ -involution makes A into a pre- C ∗ -algebra,and the ribbon and coboundary structure satisfies R ∈ M ( A ⊗ A ), v , w ∈ M ( A ).A (discrete) w-Hopf algebra A satisfying axioms a)–d) will be called a (discrete) Her-mitian coboundary w-Hopf algebra. Remark 21.4. a) Note that the definitions do not depend on the choice of the antipodeby Prop. 4.9. Furthermore, when A is discrete an antipode may always be chosen com-muting with ∗ by Remark 11.5. In the rest of the section antipodes ( S, α, β ) will bechosen with S commuting with ∗ for discrete algebras. These antipodes are of the form xS ( ) x − , xα, βx − ) with x unitary and uniquely determined. b) The equality required ind) between the R -matrices of ˜ A and ( A op ) E amounts to R ∗− = E R E − .We discuss a simple example. Example 21.5.
Let G be a compact group and C ∞ ( G ) the Hopf ∗ -algebra of functions on G which are finite linear combinations of matrix coefficients of unitary finite dimensionalrepresentations u of G . The coproduct and antipode are defined as usual by ∆( f )( g, h ) = f ( gh ) and S ( f )( g ) = f ( g − ). Then the dual ∗ -algebra is isomorphic to Π u ∈ IrrG B ( H u ), with H u the Hilbert space of u . The algebraic direct sum A = L u ∈ IrrG B ( H u ) is a discrete Hopf ∗ -algebra with dual coproduct ˆ∆ and antipode ˆ S . We have A = A op by commutativity of C ∞ ( G ), and it follows that with the trivial R -matrix and ribbon structure, A is a discreteunitary coboundary Hopf algebra.The example gives a natural interpretation of axiom d) when A is thought of as the dualof the algebra of functions on a noncommutative space. Remark 21.6.
The relationship between the multiplier discrete algebra associated to theforgetful functor of Rep( U q ( g )) and U q ( g ) has been considered in detail by Neshveyev andTuset in Sect. 2 in [100] for q >
0, and it beautifully gives a connection between twodifferent approaches to quantum groups by Woronowicz and Drinfeld. Quite remarkablyto us, the relevance of an analogous tannakian approach for a topological description of U q ( g ) has been explained by Sawin in Sect. 1 in [114] motivated by the construction of the R -matrix.The following proposition gives a characterization of the Kac-type property for an an-tipode, see Def. 8.19. Proposition 21.7.
Let A be a w-Hopf algebra with a ∗ -involution making it into a ∗ -algebra, strong antipode S such that ∆ op ( a ) ∗ = ∆ op ( I ) ∗ ∆( a ∗ )∆ op ( I ) ∗ for all a ∈ A . Then S commutes with ∗ (thus is of Kac type) if and only if P i a i S ( b ∗ i ) ∗ = I , where ∆( I ) = P i a i ⊗ b i . This is always the case when ∆ op ( I ) ∗ = ∆( I ) , that is when A is a Hermitiancoboundary w-Hopf algebra with compatible ∗ -involution in the sense of Sect. 23.Proof. The necessity of the condition follows from the antipode axiom (4.7). For the suf-ficiency, note that, if S is a strong antipode then ( S, ,
1) satisfies (4.7) and by Prop. 6.5
EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 117 and its proof this equation suffices to make a triple ( S, ,
1) into an antipode, with S anantiautomorphism. Starting with our assumptions, we may slightly modify the computa-tions in the proof of Prop. 8.20 and show that ( ˜ S, ,
1) satisfies (4.7), with ˜ S ( a ) = S ( a ∗ ) ∗ ,thus this is another strong antipode, and the proof is completed by uniqueness of a strongantipode. (cid:3) Given any central invertible element z ∈ A with ε ( z ) = 1 we setΘ z := z − ⊗ z − ∆( z ) , R z := R Θ z . (Note that when A is a w-bialgebra, E is necessarily a 2-cocycle of A op by Prop. 6.13.Similarly, Θ z and R z are 2-cocycles by Prop. 6.17.) We have A Θ z = A as quasitriangularweak quasi bialgebras thanks to centrality of z and since the twisting operation can beperformed in stages, ( A F ) G = A GF , we see that R and R z twist A in the same way.Therefore A R z = A R = A op . Furthermore the deformed R -matrix yields a generalised coboundary on Rep( A ) via Σ ρ ⊗ σ ( R z ). We set R = R Θ w . (21.1)The element R first introduced by Drinfeld in his work on quasi-Hopf algebras [34], is thealgebraic analogue of the coboundary symmetry considered in Sect. 20. Proposition 21.8.
The twist R satisfies R R = ∆( I ) . Therefore Σ ρ ⊗ σ ( R ) ∈ ( ρ ⊗ σ, σ ⊗ ρ ) is a coboundary of Rep( A ) .Proof. We have R R = R w − ⊗ w − ∆ op ( w ) Rw − ⊗ w − ∆( w ) = R Rw − ⊗ w − ∆( w ) = R Rv − ⊗ v − ∆( v ) = ∆( I ) . (cid:3) By axiom d), the element E is required to be a trivial twist from A op to ˜ A . It followsthat ˜ A = ( A op ) E = ( A R ) E = A ER , hence˜ A = A Ω , Ω = ER (21.2)as quasitriangular weak quasi-bialgebras. Theorem 21.9.
Let A be a (discrete) Hermitian coboundary weak quasi-Hopf algebra.Then A is Ω -involutive with Ω = ER = ER Θ w . Furthermore the induced braided symmetry Σ ρ ⊗ σ ( R ) ∈ ( ρ ⊗ σ, σ ⊗ ρ ) , and therefore coboundary symmetry Σ ρ ⊗ σ ( R ) ∈ ( ρ ⊗ σ, σ ⊗ ρ ) are unitary in Rep h ( A ) .Proof. We need to show that Ω is selfadjoint. By construction, ∆( I ) and ∆( I ) ∗ are re-spectively domain and range of Ω. The R -matrices of ˜ A and A op are respectively given by
18 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI ˜ R = R ∗− and R op = R thanks to Prop. 7.2. Equality between the R -matrices of ˜ A and( A op ) E gives R ∗− = E R E − , hence R ∗ = ER − E − . We may write Ω in the formΩ = ER Θ w = ERw − ⊗ w − ∆( w ) = Ew − ⊗ w − ∆ op ( w ) R = w − ⊗ w − E ∆ op ( w ) R = w − ⊗ w − ˜∆( w ) ER.
We also have E − E ∗ = ( E − ) E ∗ = (∆ op ( I )∆( I ) ∗ ) ∆ op ( I ) ∗ ∆( I ) = ∆( I ) . Hence Ω ∗ = R ∗ E ∗ ∆( w − ) w ⊗ w = ER − E − E ∗ ∆( w − ) w ⊗ w = ER − ∆( w − ) w ⊗ w = ER ( R R ) − ∆( w − ) w ⊗ w = ER ∆( w ) w − ⊗ w − = ER Θ w = Ω . Unitarity of the braided symmetry follows from the property that ˜ A = A Ω as quasi-triangular weak quasi-bialgebras and Prop. 10.2. (cid:3) In the rest of this section we endow A with the Ω-involution Ω = ER . Note that theHermitian form on the tensor product of two representations associated to Ω is given by( ζ , ζ ′ ) Ω = ( ζ , Rζ ′ ) . (21.3) Remark 21.10.
We may interpret the trivial twist E as follows. It is non-trivial preciselywhen R is not selfadjoint. This follows from the equation Ω = Ω ∗ . The subclass ofHermitian coboundary wqh for which R is already selfadjoint will be considered moreclosely in the next section.We discuss how to construct examples of Hermitian coboundary weak quasi-Hopf alge-bras with strongly trivial Ω-involution in the sense of Defn. 8.9. The following examplereduces the problem to the construction of Hermitian coboundary weak quasi-Hopf alge-bras with trivial Ω-involution. The step of constructing a unitary coboundary with trivialΩ-involution will be considered in the next section. Proposition 21.11.
Let A be a Hermitian coboundary weak quasi-Hopf algebra with trivialinvolution Ω = ∆( I ) ∗ ∆( I ) , Ω − = ∆( I )∆( I ) ∗ . Then the twist T (or T ′ ) defined in Remark8.10 making the Ω -involution strongly trivial turns A into another Hermitian coboundaryweak quasi-Hopf algebra A T .Proof. Notice that axioms a)–c) are invariant under any twist. For axiom d), we havethat T ∗ T = E , and it easily follows that the twist E T = ∆ T ( I ) ∗ ∆ op T ( I ) has inverse E − T = ∆ op T ( I )∆ T ( I ) ∗ . Moreover f A T = ( ˜ A ) T − ∗ and ( A T ) op = ( A op ) T , it follows thataxiom d) for A T is equivalent to ˜ A = ( A op ) T ∗ E T T . We have T ∗ E T T = E , thus axiom d)holds for A T also. (cid:3) EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 119
Definition 21.12.
Let A be a (discrete), Hermitian coboundary weak quasi Hopf (w-Hopf) algebra. If Ω = ER Θ w is positive in A ⊗ A ( M ( A ⊗ A )) then A will be called a unitary (discrete), coboundary weak quasi-Hopf (w-Hopf) algebra.Recall [111, 122] that an Hermitian (unitary) ribbon category is a *-category (C*-category) C equipped with a right duality ( ρ ∨ , b ρ , d ρ ), unitary braided symmetry ε ( ρ, σ )and unitary ribbon structure v ∈ (1 ,
1) such that b ∗ ρ = d ρ ◦ ε ( ρ, ρ ∨ ) ◦ v − ρ ⊗ ρ ∨ d ∗ ρ = 1 ρ ∨ ⊗ v ρ ◦ ε ( ρ ∨ , ρ ) − ◦ b ρ . (21.4)It follows from (20.15) that the quantum dimension in a Hermitian ribbon category maybe computed as d ( ρ ) = d ρ d ∗ ρ = b ∗ ρ b ρ . Theorem 21.13.
Let A be a Hermitian (unitary) coboundary weak quasi-Hopf algebrawith an antipode ( S, α, β ) such that S commutes with ∗ . Then Rep h ( A ) (Rep + ( A )) is aHermitian (unitary) ribbon category with the canonical duality ( ρ ∨ = ρ c , b ρ , d ρ ) associatedto A as in Example 11.6 if and only if β = α ∗ . This equation holds if A is discrete andadmits an antipode of Kac type.Proof. We need to give a right duality ( ρ ∨ , b ρ , d ρ ) satisfying (21.4). We show that thisholds for the canonical duality ( ρ ∨ = ρ c , b ρ , d ρ ) associated to A as in Example 11.6 and afixed antipode ( S, α, β ) such that S commutes with ∗ . We only verify the equation on theright in (21.4). We have d ∗ ρ = r ρ = Ω − X µ i e i ⊗ α ∗ e i , b ρ = X i βe i ⊗ µ i e i , with e i an orthonormal basis. A computation gives for a , b ∈ A , a ⊗ b X µ i e i ⊗ α ∗ e i = X µ i e i ⊗ bα ∗ S ( a ) e i . Taking into account S ( w ) = w , ε ( w ) = 1, and the antipode property (4.7), it follows that∆( w ) ∗ w ⊗ w X µ i e i ⊗ α ∗ e i = X µ i e i ⊗ α ∗ ve i . On the other hand, Ω − = R − ∆( w ) ∗ w ⊗ w . It follows that d ∗ ρ = Ω − X µ i e i ⊗ α ∗ e i = 1 ⊗ v ρ R − X µ i e i ⊗ α ∗ e i =1 ⊗ v ρ ◦ ε ( ρ ∨ , ρ ) − ◦ X α ∗ e i ⊗ µ i e i . Thus the equation on the right in (21.4) holds if and only if β = α ∗ . (cid:3) We next identify the element ω defined in Prop. 8.16 with the element associated to thespherical structure on Rep( A ), as in Theorem 20.18, see also Cor. 7.11, in the importantspecial case of antipode of Kac type, see Def. 8.19. Proposition 21.14.
Let A be an Hermitian coboundary weak quasi-Hopf algebra withantipode S of Kac type. Then ω = uv − where u is Drinfeld element associated to S introduced in Definition 7.6.
20 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI
Proof.
Since A has a strong antipode S , ω = m ◦ S ⊗ − ) by (8.9). With the samenotation as in Def. 7.6, we have Ω − = R − ∆( w ) ∗ w ⊗ w = P j r j w ∗ w ⊗ t j w ∗ w . Recall fromProp. 7.7 that S is the inner automorphism induced by u and that u − = P j S − ( t j ) r j .It follows that ω = P j S ( r j w ∗ w ) t j w ∗ w = S ( S − ( t j w ∗ w ) r j w ∗ w ) = S ( S − ( w ∗ w ) u − w ∗ w ) = S ( u − S ( w ∗ w ) w ∗ w ) = S ( u − v ) , for the last equality we have used axiom c) of Def. 21.1.On the other hand, S ( u − v ) = uv − by Remark 7.10. (cid:3) Let A be an Hermitian coboundary weak quasi-Hopf algebra. Replacing the choice of w with another square root w ′ of v satisfying the properties in c) of Def. 21.1 gives rise toanother Hermitian coboundary weak quasi-Hopf algebra with the same the same structureas A and new square root of the ribbon element given by w ′ , and correspondingly a newΩ w ′ , and therefore a new tensor ∗ -category, denoted Rep ′ h ( A ). We may write w ′ = wy with y a (unitary) central square root y of I in M ( A ) satisfying c), that is ε ( y ) = 1 and S ( y ) = y . Conversely, any y ∈ M ( A ) with these properties arises in this way. The newΩ w ′ differs from Θ w by the 2-coboundary Θ y = y − ⊗ y − ∆( y ), that isΩ w ′ = Ω w Θ y . In particular, Ω w ′ = Ω w if and only if y is a 1-cocycle: ∆( y ) = y ⊗ y ∆( I ). Proposition 21.15.
Assume that A is discrete. a) The functor F : Rep h ( A ) → Rep ′ h ( A ) acting identically on objects and morphismswith identity natural transformation F ρ,σ is a tensor ∗ -functor and an equivalence.There is no unitary tensor ∗ -functor between these categories unless y = w ′ w − isa -cocycle. b) If A is a unitary coboundary weak quasi-Hopf algebra with respect to w and Ω w ′ ispositive with respect to some other w ′ satisfying c) in Def. 21.1 then Ω w = Ω w ′ and R w = R w ′ .Proof. a) The categories Rep h ( A ) and Rep ′ h ( A ) have the same tensor structure and thesame ∗ -category structure, and the functor F becomes the identity functor for these sub-structures, thus it is a tensor ∗ -functor and an equivalence when the natural transformation F ρ,σ : F ( ρ ) ⊗ ′ F ( σ ) → F ( ρ ⊗ σ ) acts as identity. Here we have used different symbols todenote the two different tensor products. The Hermitian form of F ( ρ ) ⊗ ′ F ( σ ) differs fromthat of F ( ρ ⊗ σ ) by the action of Θ y . On the other hand y acts as ε ρ , where ε ρ = ± ρ , and all the ε ρ determine y . In particular, Θ y at most changesthe sign of the Hermitian form of an irreducible component of F ( ρ ⊗ σ ), and if this happensthen F ( ρ ) ⊗ ′ F ( σ ) and F ( ρ ⊗ σ ) are not unitarily equivalent, by Prop. 9.9. Hence all F ρ,σ are unitary if and only if Θ y = ∆( I ). b) If two choices w and w ′ both define positiveoperators Ω w and Ω w ′ then F : Rep + (A) → Rep + ′ (A) is a tensor ∗ -equivalence betweentensor C ∗ -categories hence by Prop. 2.17 c) polar decomposition of the tensor structuregives a unitary tensor equivalence. It follows from the the previous part that y = w ′ w ∗ isa 1-cocycle, hence Ω w = Ω w ′ and also R w = R w ′ . (cid:3) EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 121
We next construct an involutive antipode for all the twists of a unitary ribbon weakquasi-Hopf algebra of Kac type under a spectrum condition.
Proposition 21.16.
Let A be a unitary coboundary weak quasi-Hopf algebra. If A has anantipode of Kac type S with associated unitary Drinfeld element u such that − / ∈ Sp( uv − ) then for any twist F of A , A F endowed with twisted involution ( ∗ , Ω F ) admits an antipode ( ˜ S, ˜ α, ˜ β ) such that ˜ S commutes with ∗ and the corresponding element as in Prop. 8.16 is ˜ ω = 1 . In particular, ˜ S = 1 .Proof. The element ω corresponding to S and A is given by ω = uv − by Prop. 21.14, whichis unitary. Let ( S, α, β ) be the twisted antipode of A F as in (4.11), so ω F = ω by Prop.8.18 b). For an invertible x , the antipode (Ad( x ) S, xα, βx − ) of A F has associated element˜ ω = xωS − ( x ) ∗ by Prop. 8.18 a). We set x − = ω / , the continuous functional calculusof the principal branch of the square root function, so x is unitary. Since S ( ω ) = ω − itfollows that S ( x ) = x − and therefore ˜ ω = 1. (cid:3) We shall see that the weak quasi-Hopf algebras arising from VOAs as satisfying theassumptions of Sect. 17 have a natural involutive antipode commuting with ∗ .22. A categorical characterization of discrete hermitian coboundarywqh
When we start with a discrete Hermitian (unitary) coboundary weak quasi-Hopf algebra A then the C*-structure of A gives rise to the linear C*-category C + = Rep + ( A ) of Hilbertspace representations of A . We also have the tensor category C = Rep( A ) of vector spacerepresentations of A which has additional structure, the braiding, the ribbon structureand coboundary symmetry. We next give a categorical description of the construction ofthe Hermitian (unitary) structure of a Hermitian or unitary coboundary weak quasi-Hopfalgebras. For simplicity, we discuss a detailed proof only in the unitary case.We recall that ribbon and coboundary structures in tensor categories have been studiedin Sect. 20. In particular, by Theorem 20.21 a coboundary symmetry c w may be associatedto a ribbon category C with braided symmetry c , ribbon structure v ∈ (1 ,
1) when thereis a natural isomorphism w ∈ (1 ,
1) compatible with duality which is a square root of v .In the setting of unitary categories when c and w are unitary then c w is unitary, and alsoselfadjoint as ( c w ) = 1. Theorem 22.1.
Let ( C , ⊗ , α, c, v ) be a ribbon category, w ∈ (1 , a square root of v compatible with duality, C + a semisimple C ∗ -category and let F : C + → C be a linearequivalence. Let ( G , F, G ) : C → Herm ( C → Hilb ) be a faithful weak quasitensor functorwith symmetric dimension function such that G + = GF : C + → Herm ( C + → Hilb ) isa ∗ -functor. Then the discrete pre- C ∗ -algebra A + = Nat ( G + ) endowed with the naturalribbon weak quasi Hopf algebra structure and Ω -involution induced by duality becomes anHermitian (unitary) coboundary weak quasi-Hopf algebra if and only if G ( α ) , G ( c ) , and G ( v ) are unitary and moreover ( F, G ) satisfies the following conditions F σ,ρ Σ( ρ, σ ) F ∗ ρ,σ = G ( c w ( ρ, σ )) (22.1)
22 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI G ∗ σ,ρ Σ( ρ, σ ) G ρ,σ = G ( c w ( σ, ρ ) − ) (22.2) with Σ the permutation symmetry of Herm ( Hilb ). In this case, C + becomes an Hermitian(unitary) ribbon tensor category and F : C + → C a ribbon tensor equivalence.Moreover there is a unitary ribbon tensor equivalence E : C + → Rep h ( A ) ( C + → Rep + ( A ) ) preserving the coboundary structures such that F A E ≃ G unitarily monoidally,with F A the forgetful functor of Rep + ( A ) .Proof. By theorem 5.6, A = Nat ( G ) becomes a ribbon weak quasi-Hopf algebra withcoproduct and associator defined by ( F, G ) and R -matrix R defined in the proof andribbon structure G ( v ρ ). We transfer this structure to A + via the isomorphism A → A + induced by F . By Prop. 19.4, A + becomes naturally a unitary weak quasi-bialgebra if andonly if G ( α ) is unitary. The Ω-involution of A + is given by Ω = ˜ F ∗ ˜ F and Ω − = ˜ G ˜ G ∗ respectively, where ˜ F and ˜ G correspond to F and G via the isomorphism, as in the proof ofProp. 19.4. We have ˜ A + = A +Ω as weak quasi-bialgebras. When G ( v ρ ) is unitary then thenatural transformation G ( w F ( x )) ) defines a unitary square root of the ribbon structure of A + , and axioms a), b), c) of Def. 21.1 hold. We show that with this structure axiom d) isequivalent to (22.1) and (22.2) if G ( c ) is unitary. Note that A op = ( A ) R as quasitriangularweak quasi-bialgebras. It follows that d) may equivalently be formulated as ˜ A + = ( A + ) ER as quasitriangular weak quasi-bialgebras together with the requirement that E is a trivialtwist, that is E − = E ′ , where E = ∆( I ) ∗ ∆ op ( I ) and E ′ = ∆ op ( I )∆( I ) ∗ . On the otherhand, equations (22.1) and (22.2) are respectively equivalent toΣ G σ,ρ F σ,ρ Σ F ∗ ρ,σ F ρ,σ = Σ G σ,ρ G ( c w ( ρ, σ )) F ρ,σ , (22.3) G σ,ρ G ∗ σ,ρ Σ G ρ,σ F ρ,σ Σ = G σ,ρ G ( c w ( σ, ρ ) − ) F ρ,σ Σ . (22.4)We know that R and R − correspond to Σ G σ,ρ G ( c ( ρ, σ )) F ρ,σ and G σ,ρ G ( c ( σ, ρ ) − ) F ρ,σ Σ. Itfollows from a computation that R and R − in turn correspond to Σ G σ,ρ G ( c w ( ρ, σ )) F ρ,σ and G σ,ρ G ( c w ( σ, ρ ) − ) F ρ,σ Σ. It follows that equations (22.3) and (22.4) are in turn equivalentto E ′ Ω = R , Ω − E = R − , in other words E ′ = E − and Ω = ER . On the otherhand, the R -matrices of ˜ A and A Ω coincide by Prop. 10.2 as G ( c ) is a unitary braidedsymmetry. Thus the proof of axiom d) is complete. Conversely, when A is a unitarycoboundary weak quasi-Hopf algebra and C = Rep( A ) then the natural weak quasi-tensorstructure of the forgetful Rep( A ) → Hilb satisfies F ∗ ρ,σ = ρ ⊗ σ (∆( I ) ∗ )Ω ρ,σ and similarly G ∗ ρ,σ = Ω − ρ,σ ρ ⊗ σ (∆( I ) ∗ ). Moreover c w corresponds to Σ R . It follows that the unitaritystatements and (22.1) and (22.2) are verified. The property that C + is an Hermitian(unitary) ribbon category follows from Theorem 21.13. In the unitary case it also followsthat the canonical tensor equivalence E described in Theorem 5.6 is unitary by Prop. 19.4,see also Theorem 10.5 and preserves the coboundary symmetries by construction. (cid:3) Remark 22.2. a) It follows from the proof of Theorem 22.1 and that of Theorem 19.2 thatwhen C has a generating object (i.e. its powers contain every irreducible as a subobject)then Theorem 22.1 holds if equations (22.1) and (22.2) are known to hold only for pairs ρ , σ such that one of them, say ρ , is the generating object and the other varies among theirreducible objects of C , or alternatively among the choice of a tensor power ρ n for each EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 123 integer n . b) Equations (22.1) and (22.2) link the coboundary symmetry c w of C to thepermutation symmetry of Hilb through the weak quasi-tensor structure of F . In particular,when F is a unitary quasi-tensor functor, c is a permutation symmetry and v = w = 1then we recover the notion of symmetric functor. c) It follows from Theorem 22.1 andProp. 21.15 b) that c w does not depend on the choice of w . Remark 22.3.
It follows that Theorem 21.13 admits a categorical formulation as well.Indeed, we may define a discrete coboundary weak quasi-Hopf algebra A equivalently asa semisimple ribbon tensor category ( C , ⊗ , α, c, v ) endowed with a square root w of theribbon element v compatible with duality and the structure of a C ∗ -category with a weakquasi-tensor faithful functor ( G , F, G ) such that all the conditions of Theorem 22.1 hold(we are choosing C + = C and F identity). Then any right duality ( ρ ∨ , b ρ , d ρ ) is of the formdescribed in Example 11.6 by the proof of Theorem 5.6 (d), as an antipode ( S, α, β ) mayalways be chosen such that S commutes with ∗ by Remark 11.5. Thus by Theorem 21.13,the condition β = α ∗ is equivalent to the compatibility equations (21.4) making C into anHermitian ribbon category with respect to ( c, v, ρ ∨ , b ρ , d ρ ).Taking into account the historical motivation briefly discussed in the introductory part ofSection 21, we are led to look for special examples with R -matrix given by a weak analogueof a 2-coboundary. The next remark shows that the construction of these examples isrelated to the study of unitary structures, having a suitable triviality property. Remark 22.4.
If an Hermitian coboundary A has trivial involution as introduced in Def.8.9 then by definition Ω is a trivial twist, thus we have from relation (21.2)Ω = ∆( I ) ∗ ∆( I ) , R = ∆ op ( I )∆( I ) ∗ w ⊗ w ∆( w − ) , R = ∆ op ( I )∆( I ) ∗ ∆( I ) . (22.5)Conversely, if A is Hermitian coboundary and the R -matrix takes the previous form thennecessarily the involution is trivial. We have a particular case, when the Ω-involution of A is strongly trivial (Ω = Ω − = ∆( I )) thenΩ = ∆( I ) , R = ∆ op ( I ) w ⊗ w ∆( w − ) , R = ∆ op ( I )∆( I ) . (22.6)We recall from Example 21.11 that strongly trivial Ω-involutions can be obtained fromtrivial Ω-involutions via suitable twisting. Moreover, when A is in addition unitary discretethan any trivial Ω-involution is strongly trivial by Prop. 2.16.In the next section we consider the question of constructing new examples of unitarycoboundary weak quasi-Hopf algebras with such triviality properties from old ones, andTheorem 22.1 will turn out useful. To construct such examples, we look for twist defor-mation of given examples that respect the structure, that may perhaps be regarded asan abstract analytic analogue of part of the arguments involved Drinfeld-Kohno theoremfollowing [34].23. Compatible unitary coboundary wqh and an abstract analogue ofDrinfeld-Kohno theorem
We know from a theorem of Galindo [49] that a braiding of a unitary fusion categoryis always unitary. Now we reverse the question and ask is there a way of constructing a
24 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI unitary braided tensor category with a unitary braiding, equivalent to a given a braidedsemisimple tensor category C ? In applications we may already have a linear C ∗ -category C + and a linear equivalence F : C + → C and we want to turn C + into a unitary braidedtensor category. Furthermore, if we have two braided tensor categories C and C whichare linearly equivalent to the same C ∗ -category C + via F i : C + → C i , i = 1, 2, under whatcircumstances the corresponding constructions give unitarily equivalent braided tensorcategories C and C ? If this can be achieved, it will follow in particular that C and C are also equivalent as braided tensor categories. In this section we set up a specific situationand we construct a unitary braided tensor quasi-equivalence ( E , E ) : C → C . In otherwords we reduce the problem to verification of the equation concerning the associativitymorphisms only, that is equation (2.6), (2.7) with ( E , E , E − ) in place of ( F , F , G ). Indoing this, we follow ideas of Drinfeld [34] in his work on Drinfeld-Kohno theorem, exceptfor as already said we forget the associativity morphisms, and again ideas of Wenzl [128]in his work of the unitary structures of fusion categories C ( g , q, ℓ ) of quantum groups atroots of unity.In the introduction of Section 21 we have interpreted axiom d) of Definition 21.1 as anoncommutativity property of the function algebra from a dual viewpoint. This interpre-tation disregards the trivial twist E , and therefore becomes more meaningful when thetrivial twist is actually trivial. This leads us to the following stronger definition. Definition 23.1.
A Hermitian coboundary weak quasi-Hopf algebra ( A, ∆ , Φ , R, v, w, ∗ , S, α, β )is called compatible with the ∗ -involution if it satisfies one of the following equivalent con-ditions,1) E = ∆( I ) ∗ = ∆ op ( I ),2) ∆( a ) ∗ = ∆ op ( a ∗ ), a ∈ A ,3) Ω = R ,4) R is selfadjoint.Thus axiom d) of Def. 21.1 is replaced by the stronger axiomd’) ˜ A = A op as quasitriangular weak quasi-bialgebras.In particular we have R ∗− = R . Unitary, discrete, or w -Hopf versions are naturallydefined.Example 21.5 is of this kind. In Sect. 24 we construct examples associated to fusion cat-egories C ( g , q, ℓ ) associated to U q ( g ) at certain roots of unity with compatible ∗ -involution.In the rest of the paper we restrict to the unitary case. Proposition 23.2.
Let A be a discrete unitary coboundary wqh with a generating repre-sentation ρ . Then A has compatible ∗ -involution if and only if σ ⊗ ρ ( R ) , σ ⊗ ρ ( R − ) σ ⊗ ρ ⊗ ρ ( I ⊗ R ⊗ ∆( R )) , ρ ⊗ ρ ⊗ σ ( R ⊗ I ∆ ⊗ R )) are positive for every irreducible representation σ . EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 125
Proof.
Necessity is clear. We note that the associativity morphisms σ ⊗ ρ ⊗ ρ (Φ) and ρ ⊗ ρ ⊗ σ (Φ) are unitary w.r.t. the given unitary coboundary structure, which is definedby R on the involved subspaces. By Theorem 19.2, A becomes a unitary coboundarywqh with compatible ∗ -involution. On the other hand the original coboundary structureΩ = ∆( I ) ∗ R and the new compatible coboundary structure Ω ′ = R coincide of the spacesof σ ⊗ ρ and therefore coincide everywhere by the conclusion of Theorem 19.2. (cid:3) The following remark is an analogue of Remark 22.4 for the subclass of wqh of thissection, and takes a perhaps remarkable stronger form that seems to remind of the formtaken by the R -matrix in the specific case of Drinfeld category [34] for quasi-Hopf algebras. Remark 23.3.
Let A be an Hermitian coboundary wqh with compatible ∗ -involution andtrivial Ω-involution. Then we haveΩ = ∆( I ) ∗ ∆( I ) = R, R = ∆( I ) ∗ w ⊗ w ∆( w − ) . (23.1)When the Ω-involution is in addition strongly trivial (recall that this is automatic when A is discrete unitary by Prop. 2.16) thenΩ = ∆( I ) = R, R = w ⊗ w ∆( w − ) . (23.2)In particular, A has a cocommutative coproduct (∆ = ∆ (op) ) by centrality of w .Note that if T is a twist of A with left inverse T − then by definition ∆( I ) is the domainof T and range of T − . If A has a compatible ∗ -involution then we also have that T − ∗ has domain ∆ op ( I ) ∗ = ∆( I ), T has domain ∆ op ( I ) = ∆( I ) ∗ , T ∗ has range ∆( I ) ∗ .The following result is our abstract analogue of Drinfeld-Kohno theorem. Theorem 23.4.
Let A = ( A, ∆ , Φ , R, v, w ) be a discrete unitary coboundary weak quasi-Hopf algebra with compatible ∗ -involution ( ∗ , Ω = R ≥ . Let ( T, T − , P, Q ) a quadrupleof elements in M ( A ⊗ A ) : such that T is a twist of A with left inverse T − , P , Q areselfadjoint projections in M ( A ⊗ A ) such that P Q = 0 , P + Q = IT = ( T − ) ∗ , R = T ∗ T − , R − = T − − ( T − ) ∗ . where T − = ( P − Q ) T, T − − = T − ( P − Q ) . Then a) A T is another discrete unitary coboundary weak quasi-Hopf algebra with compatibleinvolution, having a similar structure with respect to the quadruple (∆ T ( I ) , ∆ T ( I ) , P, Q ) .Therefore the twisted structure of A T is given by Ω T = ∆ ∗ T ( I )∆ T ( I ) − = ∆ op T ( I )∆ T ( I ) − = R T ≥ R T = ∆ op T ( I )∆ T ( I ) − w ⊗ w ∆ T ( w − ) , where ∆ T ( I ) − = ( P − Q )∆ T ( I ) ,
26 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI b) If ρ and σ are two Hilbert space ∗ -representations of A such that ρ ⊗ σ ( QT ) = 0 and ρ ⊗ σ ( T − Q ) = 0 then ρ ⊗ σ ( T − ) = ρ ⊗ σ ( T ) , ρ ⊗ σ (( T − ) − ) = ρ ⊗ σ ( T − ) . Moreover, ( F T ) ρ,σ ( F T ) ∗ ρ,σ = 1 , ( G T ) ∗ ρ,σ ( G T ) ρ,σ = 1 , ( G T ) ρ,σ = ( F T ) ∗ ρ,σ . c) If the assumptions in b) holds for any pair of irreducible ∗ -representations ρ , σ of A then T = T − , T − = ( T − ) − , ∆ T ( I ) = ∆ T ( I ) − . Moreover the twisted structure ( F T , G T ) is strongly unitary. Thus the R -matrix R T and the hermitian form Ω T simplify further as in (23.2). In particular, thecoproduct ∆ T of A T is cocommutative, ∆ T = ∆ op T .Proof. a) We have T − ( T − ) ∗ = ∆( I ), T ∗ T = ∆( I ) ∗ . Let (
F, G ) be the weak quasi-tensorstructure defining the forgetful functor of A . Then by Theorem 22.1 equations (22.1) and(22.2) hold for ( F, G ). Let ( F T , G T ) be the new weak quasi-tensor structure obtained fromthe twist T , F T = F T − , G T = T G . We have F T Σ F ∗ T = F T − Σ( T − ) ∗ F ∗ = F T − ( T − ) ∗ Σ F ∗ = F ( GF )Σ F ∗ = F Σ F ∗ and similarly G ∗ T Σ G T = G ∗ T ∗ T Σ G = G ∗ ( GF ) ∗ Σ G = G ∗ Σ G. It follows that equations (22.1) and (22.2) hold for ( F T , G T ). The twisted R -matrix R T induces a unitary braided symmetry in Rep + ( A T ) by Remark 10.3. Moreover the twistedassociator of A T is unitary Rep + ( A T ) by invariance of axioms of Ω-involution under twist-ing. It follows from Theorem 22.1 again that A T is a unitary coboundary weak quasi-Hopfalgebra. It follows from Prop. 7.2 c) that R T = T RT − = T T ∗ ( P − Q ) T T − = T ( T − ) ( P − Q ) T T − = ∆ op T ( I )( P − Q )∆ T ( I ) . We also have ∆ T ( I ) ∗ = ( T − ) ∗ T ∗ = T T − = ∆ op T ( I ) thus A T has a compatible ∗ -involution. This is also equivalent to R T = Ω T . The formula for R T follows fromthe definition of R in (21.1) for a general hermitian coboundary wqh. b) In this casewe have ρ ⊗ σ ( T − ) = ρ ⊗ σ ( P T ) = ρ ⊗ σ (( P + Q ) T ) = ρ ⊗ σ ( T ). In a similar way, ρ ⊗ σ ( T − − ) = ρ ⊗ σ ( T − ). It follows that ( F T ) ρ,σ ( F T ) ∗ ρ,σ = F ρ,σ ρ ⊗ σ ( T − ( T − ) ∗ ) F ∗ ρ,σ = F ρ,σ ρ ⊗ σ ( R ) − F ∗ ρ,σ = F ρ,σ G ρ,σ G ∗ ρ,σ F ∗ ρ,σ = 1. One similarly shows that ( G T ) ∗ ρ,σ ( G T ) ρ,σ = 1.The equality ( G T ) ∗ ρ,σ = F ρ,σ follows from Prop. 2.16. c) This follows from b) and Tannaka-Krein duality. (cid:3) EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 127
Compatible unitary coboundary w-Hopf algebras A W ( g , q, ℓ ) as asubquotient of U q ( g )In this section we identify a natural functor W : C ( g , q, ℓ ) → Vec associated to the samedimension function D as in Sect. 16, and thus it is a particular case of the former.To do this, we consider the tensor structure of C ( g , q, ℓ ) ( q not necessarily minimal) of[128]. This gives rise to the mentioned forgetful functor W , and we introduce a weak tensorstructure on W , and in this way we have a canonical w-Hopf algebra A W ( g , q, ℓ ).When q is a minimal root, the work of [128] shows that U q ( g ) is a Hermitian coboundaryHopf algebra with compatible involution (in a topological sense), we review this result inTheorem 24.1. This Hermitian structure underlies the unitary structure of C + ( g , q, ℓ ). Inconclusion, we have an epimorphism of ∗ -algebras U q ( g ) → A W ( g , q, ℓ )and the unitary coboundary structure of A W ( g , q, ℓ ) arises naturally from the (non-semisimple)hermitian coboundary structure of U q ( g ) through Tannakian reconstruction of the unitarycoboundary structure of C + ( g , q, ℓ ).The weak tensor structure is not unique but when we change it then the w-Hopf algebrachanges only by a trivial twist. The special case g = s l N will be useful for the constructionof tensor equivalences studied in Sect. 18. We also note that in this case we recover theexample constructed in [23] with a different method.Recall that the algebra U q ( g ) at complex roots of unity was introduced in Sect. 16,and we assume the same setting as there. In particular, it becomes a ribbon complexHopf algebra with a ∗ –involution, and is topological in the sense of [114]. Note that the R -matrix R and the ribbon element v ∈ U q ( g ) depend only on the choice of q /L , see Sect.1 in [114], Sect. 1.4 in [128]. Furthermore, a square root w ∈ U q ( g ) of v is well defined upto a sign choice in every representation entering the definition of U q ( g ), we refer to Sect.1 in [114] for details. We may summarize properties of the structure of U q ( g ) as follows. Theorem 24.1.
Let q /L be a fixed L -th root of q and consider the associated ribbonstructure ( R, v ) on U q ( g ) . Then for every square root w of v , U q ( g ) endowed with the natural ∗ -involution becomes a (topological) Hermitian coboundary Hopf algebra with compatibleinvolution and antipode of Kac type.Proof. The Kac-type property of the antipode follows from properties (16.1)–(16.2) thatstill hold for U q ( g ). Axioms b), c), d) of Def. 21.1 are shown in Lemma 1.4.1 of [128]. (cid:3) Recall also that the quotient category C ( g , q, ℓ ) was outlined in Sect. 16. We assume ℓ ′ < ∞ . Remark 24.2.
By Lemma 1.1 in [51], composition of inclusion T → T ( g , q, ℓ ) with pro-jection T ( g , q, ℓ ) → C ( g , q, ℓ ) is an equivalence of linear categories. Hence T becomesa semisimple tensor category tensor equivalent to C ( g , q, ℓ ). In the following subsection
28 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI we shall construct among other things a specific tensor structure of a certain interest-ing equivalent full subcategory G q ⊂ T making G q → C ( g , q, ℓ ) an equivalence of tensorcategories.For λ , µ ∈ Λ ℓ , one can decompose V λ ⊗ V µ uniquely up to isomorphism in T ( g , q, ℓ ), V λ ⊗ V µ ≃ ⊕ ν ∈ Λ ℓ m νλ,µ V ν ⊕ N, with N negligible. Then in C ( g , q, ℓ ), V λ ⊗ V µ ≃ ⊕ ν ∈ Λ ℓ m νλ,µ V ν . Notice that this decomposition of V λ ⊗ V ν is unique up to isomorphism but not canonical(cf. [128], and also Sect. 11.3C in [22] and references therein.)The ribbon structure of C ( g , q, ℓ ) is induced by that of the tilting category. Also theformulas for the fusion coefficients and quantum dimensions are well-known, and regulatedby the affine Weyl group, as in Sect. 2, 5 of [114], but we shall only need them in somespecial cases later on, so we refrain from recalling them in full generality. However, it willbe important for us to recall that C ( g , q, ℓ ) depends on q but the Grothendieck semiring R ( C ( g , q, ℓ )) depends only on ℓ . We shall refer to R ( C ( g , q, ℓ )) as the Verlinde fusion ring. Further properties of modularity C ( g , q, ℓ ) depend onon the choice of q /L as a primitive root of unity of order ℓ ′ L and on the order ℓ ′ of q . Werefer to the papers by Rowell and Sawin [111, 114] for a detailed treatment. For examplethe cases where 2 d | ℓ ′ give modular categories and this is the case of most physical interest,and also that meeting the purpose of our paper.More in particular, we shall mostly be interested in the“minimal roots” q = e iπ/ℓ , q /L = e iπ/ℓL , d | ℓ. Indeed in this case C ( g , q, ℓ ) is equivalent to a unitary ribbon fusion category that wedenote by C + ( g , q, ℓ ) by [128, 134], and indeed modular. A W ( g , q, ℓ ) . In this subsection q is any rootof unity of order large enough. We obtain a functor C ( g , q, ℓ ) → Vec together with a weaktensor structure (
F, G ) associated to the same dimension function D as in the previoussubsection, and correspondingly a w-Hopf algebra A W ( g , q, ℓ ). In the next subsection weconsider the case where q is a minimal root. For this construction we mostly take intoconsideration ideas in [128] that we review and extend to a general root of unity q such that ℓ is large enough in the sense of Def. 16.1. When q is a minimal root, A W ( g , q, ℓ ) becomesa unitary coboundary w-Hopf algebra. To do this, as briefly anticipated in Remark 24.2,we shall introduce a linear category G q of non-negligible tilting modules associated to afundamental representation of g . This category appears implicitly in [128]. In [23] we haveshown that G q has a natural structure of a strict (ribbon) tensor category when Vec isregarded as strict and q is a minimal root and is unitarily ribbon equivalent to C ( g , q, ℓ ).In this subsection we extend this to all roots of sufficiently large order and moreover weshall define a functor W : G q → Vec and then introduce a weak tensor structure on W that corresponds by Tannakian reconstruction to A W ( g , q, ℓ ). EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 129
Following Sect. 3.5 in [128], we choose a fundamental representation V of the Lie algebrafor each Lie type. This representation satisfies the following properties. It is irreducible if g is not of type D , and is the sum of the two half spin representations in the type D case;every irreducible of g is a subrepresentation of a power of V ; the dominant weight of V (or of each summand in type D ) lies in Λ ℓ . We consider the associated Weyl module of U q ( g ) denoted in the same way. For g = E and for all λ ∈ Λ ℓ , V λ ⊗ V decomposes into adirect sum of irreducible representations V γ with the property that the dominant weights γ appearing in the decomposition into indecomposable tilting modules T γ at most lie inΛ ℓ , thus T γ = V γ for all γ and the decomposition is completely reducibile. If furthermore g = F then the decomposition is multiplicity free, while for g = F multiplicity may arisefor γ = λ . For g = E , the summand T γ may not lie in Λ ℓ for γ = λ + κ with κ thedominant weight of V . Multiplicity may arise for γ = λ in this case also.For every f.d. simple complex Lie algebra g , an orthogonal decomposition V λ ⊗ V = ⊕ γ T γ ⊗ C m γ with T γ indecomposable tilting modules is constructed in [128]. Here the most delicatecases are F and especially E . In our understanding, these constructions hold for anyprimitive root q such that the order ℓ of q is large enough. We denote by p λ,γ : V λ ⊗ V → V γ ⊗ C m γ , λ, γ ∈ Λ ℓ the corresponding idempotent.We define the projection p := P γ ∈ Λ ℓ p κ,γ and set V ⊗ V = p V ⊗ V . We use p λ,γ toiteratively define projections p n : V ⊗ n → V ⊗ n onto the maximal non-negligible submodule V ⊗ n induced by the decomposition of V ⊗ n − ⊗ V . Remark 24.3.
By the iterative argument in the construction, every representation V ⊗ n has a canonical decomposition into irreducible subrepresentations V ( n ) γ,j , where γ denotesthe highest weight of V ( n ) γ,j and j counts the multiplicity up to isomorphism. Definition 24.4.
Let G q denote the completion with idempotents and direct sums of thefull linear subcategory of T ( g , q, ℓ ) with objects the truncated tensor powers V ⊗ n .Thus by construction G q is a semisimple linear category, that we regard it as an abstractcategory. Let Vec be realized as a strict tensor category. We regard V ⊗ n as a summandon V ⊗ n via p n , and identify the morphism space ( V ⊗ m , V ⊗ n ) in G q with the subspace ofmorphisms T ∈ ( V ⊗ m , V ⊗ n ) in T ( g , q, ℓ ) satisfying T p m = p n T = T . We set V ⊗ m ⊗ V ⊗ n := V ⊗ m + n , (24.1) S ⊗ T := p m ′ + n ′ ◦ S ⊗ T ◦ p m + n , S ∈ ( V ⊗ m , V ⊗ m ′ ) , T ∈ ( V ⊗ n , V ⊗ n ′ ) . (24.2)It is known that with this tensor product and trivial associativity morphisms G q becomesa strict ribbon tensor category equivalent to C ( g , q, ℓ ), see Theorem 5.4 in [23] for a proof.We next introduce a concrete version of G q . Definition 24.5.
Let ˜ G q denote the full representation subcategory of T ( g , q, ℓ ) with ob-jects representations which are finite direct sums of summands of the representations V ⊗ n .
30 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI
Then ˜ G q is also a linear semisimple category. There is a canonical linear equivalence˜ G q → G q taking the summand P ( n ) V ⊗ n of V ⊗ n defined by an idempotent P ( n ) regarded as an objectof ˜ G q to P ( n ) regarded as an object of G q and acting trivially on morphisms. We shall make˜ G q into a tensor category with a tensor structure (˜ G q , ⊠ , α ) in such a way that E becomesa tensor equivalence ( E , E ).Let then W : ˜ G q → Vecbe the forgetful functor. To define a tensor structure on ˜ G q we first define linear maps( F λ,µ , G λ,µ ) on W defined on pairs ( λ, µ ) ∈ Λ ℓ × Λ ℓ that will correspond to a tensorstructure ( ⊠ , α ) of ˜ G q and subsequently also to a weak tensor structure for W .For every λ ∈ Λ ℓ choose an integer n λ such that λ appears as the dominant weight ofa summand V λ of V ⊗ n λ as observed in Remark 24.3. Let p λ : V ⊗ n λ → V λ denote thecorresponding idempotent onto V λ for each λ ∈ Λ ℓ . In the following formulae we extend p λ to V ⊗ n λ in a trivial way on (1 − p n λ ) V ⊗ n λ . Proposition 24.6.
We have that p λ ⊗ p µ = p n λ + n µ p λ ⊗ p µ p n λ + n µ is a canonical idempo-tent in the semisimple category ˜ G q onto a module isomorphic to a maximal non-negligiblesubmodule of V λ ⊗ V µ in T ( g , q, ℓ ) .Proof. Notice that p λ ⊗ p µ is a morphism in T ( g , q, ℓ ) and is an idempotent by (2) in Sub-sect. 15.2 with range in the semisimple part V ⊗ ( n λ + n µ ) , thus this range is a semisimplerepresentation depending only on λ , µ up to isomorphism. If M λ,µ is a maximal idempo-tent onto a nonnegligible summand of p λ V ⊗ n λ ⊗ p µ V ⊗ n µ then p λ ⊗ p µ = p n λ + n µ M λ,µ p n λ + n µ .We have that T = p n λ + n µ M λ,µ ∈ ( M λ,µ , p n λ + n µ ) and T − = M λ,µ p n λ + n µ ∈ ( p n λ + n µ , M λ,µ )satisfy T − T = M λ,µ and T T − = p λ ⊗ p µ . (cid:3) We define V λ ⊠ V µ := p n λ ⊗ p µ V ⊗ ( n λ + n µ ) as a module of ˜ G q , thus W ( V λ ⊠ V µ ) = p n λ + n µ p λ ⊗ p µ V ⊗ ( n λ + n µ ) as a linear space. Definition 24.7.
For λ , µ ∈ Λ ℓ , let F λ,µ : V λ ⊗ V µ → V λ ⊠ V µ , G λ,µ : V λ ⊠ V µ → V λ ⊗ V µ be the morphisms in T ( g , q, ℓ ) respectively defined as the restriction of p λ ⊗ p µ = p n λ + n µ p λ ⊗ p µ p n λ + n µ to V λ ⊗ V µ and that of p λ ⊗ p µ to V λ ⊠ V µ . Thus forgetting the morphism propertywe have linear maps F λ,µ : W ( V λ ) ⊗ W ( V µ ) → W ( V λ ⊠ V µ ) , G λ,µ : W ( V λ ⊠ V µ ) → W ( V λ ) ⊗ W ( V µ ) . Proposition 24.8.
We have that F λ,µ G λ,µ = 1 and G λ,µ F λ,µ is an idempotent of T ( g , q, ℓ ) onto a maximal non-negligible submodule of V λ ⊗ V µ .Proof. The first statement is again a simple consequence of (2) in Subsect. 15.2. theremaining part follows from this and Prop. 24.9. (cid:3)
EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 131
We next extend ⊠ and F λ,µ , G λ,µ to all objects of ˜ G q . Let P ( n ) ∈ ( V ⊗ n , V ⊗ n ), P ( m ) ∈ ( V ⊗ m , V ⊗ m ) be idempotents in G q and consider morphisms describing decomposition intoirreducibles, that is S λ,j : V λ → P ( n ) V ⊗ n , S ′ λ,j : P ( n ) V ⊗ n → V λ ,T µ,k : V µ → P ( m ) V ⊗ m , T ′ µ,k : P ( m ) V ⊗ m → V µ ,S ′ λ,j S λ ′ ,j ′ = δ ( λ,j ) , ( λ ′ ,j ′ ) , X λ,j S λ,j S ′ λ,j = P ( n ) ,T ′ µ,k T µ ′ ,k ′ = δ ( µ,k ) , ( µ ′ ,k ′ ) , X µ,k T µ,k T ′ µ,k = P ( m ) . We set F P ( n ) V ⊗ n ,P ( m ) V ⊗ m = X λ,j,µ,k S λ,j ⊗ T µ,k ◦ F λ,µ ◦ S ′ λ,j ⊗ T ′ µ,k , (24.3) P ( n ) V ⊗ n ⊠ P ( m ) V ⊗ m = F P ( n ) V ⊗ n ,P ( m ) V ⊗ n ( P ( n ) V ⊗ n ⊗ P ( m ) V ⊗ m ) (24.4)and we let G P ( n ) V ⊗ n ,P ( m ) V ⊗ m be the restriction of X λ,j,µ,k S λ,j ⊗ T µ,k ◦ G λ,µ ◦ S ′ λ,j ⊗ T ′ µ,k (24.5)to P ( n ) V ⊗ n ⊠ P ( m ) V ⊗ m . Notice that F P ( n ) V ⊗ n ,P ( m ) V ⊗ m and G P ( n ) V ⊗ n ,P ( m ) V ⊗ m are indepen-dent of the choice of S λ,j , S ′ λ,j , T µ,k , T ′ µ,k by bilinearity of ⊗ . In particular, these maps andtensor products extend the previous ones on the chosen class of irreducibles. Finally, weextend this structure to any object of ˜ G q by bilinearity. Remark 24.9.
It follows that the morphism properties in Def. 24.7 and Prop. 24.8 extendto all pairs of objects of ˜ G q in place of V λ and V µ .We next define a tensor product between morphisms and associativity morphisms in ˜ G q as follows. Let ρ , σ , τ be objects of ˜ G q . For S : ρ → ρ ′ , T : σ ′ → σ ′ , set S ⊠ T = F ρ ′ ,σ ′ S ⊗ T G ρ,σ . (24.6)We endow ˜ G q with associativity morphisms α ρ,σ,τ = F ρ,σ ⊠ τ ◦ ρ ⊗ F σ,τ ◦ G ρ,σ ⊗ τ ◦ G ρ ⊠ σ,τ (24.7)Note that the maps F and G are defined up to varying the choice of the integers n λ orthe definition of the idempotents p n . Theorem 24.10.
Let g be a complex simple Lie algebra, q a complex root of unity suchthat q has order ℓ large enough. Then a) (˜ G q , ⊠ , α ) is a semisimple tensor category,
32 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI b) the canonical linear equivalence E : ˜ G q → G q admits a unique structure of tensorequivalence ( E , E ) : (˜ G q , ⊠ , α ) → ( G q , ⊗ , such that E λ,µ : E ( V λ ) ⊗ E ( V µ ) → E ( V λ ⊠ V µ ) , λ, µ ∈ Λ ℓ acts as F λ,µ and we have that E − λ,µ acts as G λ,µ , c) the pair ( F, G ) is a weak tensor structure for the forgetful functor W : ˜ G q → Vec ,therefore A W ( g , q, ℓ ) = Nat ( W ) is a ribbon w-Hopf algebra, d) a different choice of p n , p λ changes A W ( g , q, ℓ ) by a trivial twist.Proof. a) Note that S ⊠ T is composition of morphisms in T ( g , q, ℓ ) with domain and rangerepresentations of ˜ G q , thus it is a morphism in ˜ G q . By Remark 24.9, for any pair of objects ρ , σ ∈ ˜ G q , G ρ,σ F ρ,σ is an idempotent in T ( g , q, ℓ ) with range a maximal non-negligiblesummand of the tensor product tilting module ρ ⊗ σ . Thus 1 − G ρ,σ F ρ,σ is an idempotentonto the negligible summand. This observation together with property (2) in Subsect. 15.2implies that ⊠ is a bifunctor of ˜ G q . The pentagon equation can be shown again taking intoaccount property (2) and we also need (3). For example computing the short side of thepentagon equation (2.1) α ν,ρ,στ α νρ,σ,τ = F ν,ρ ( στ ) ◦ ν ⊗ F ρ,στ ◦ G ν,ρ ⊗ στ ◦ G νρ,στ F νρ,στ ◦ νρ ⊗ F σ,τ ◦ G νρ,σ ⊗ τ ◦ G ( νρ ) σ,τ we may first eliminate the central term G νρ,στ F νρ,στ , then use the commutation relation G ν,ρ ⊗ στ ◦ νρ ⊗ F σ,τ = 1 νρ ⊗ F σ,τ ◦ G ν,ρ ⊗ στ thus α ν,ρ,στ α νρ,σ,τ = F ν,ρ ( στ ) ◦ ν ⊗ F ρ,στ ◦ νρ ⊗ F σ,τ ◦ G ν,ρ ⊗ στ ◦ G νρ,σ ⊗ τ ◦ G ( νρ ) σ,τ . The computation involving the long side of the pentagon equation is slightly longer becauseof the use of ⊠ at both sides. However it can patiently be carried out and it leads toequating the left hand side. b) It is clear that E λ,µ and E − λ,µ are morphisms and are inverseof each other. Furthermore extending these morphisms by naturality to every pair ofobjects we see that they act as F and G respectively. Then we may verify the tensorialityequation (2.6) for ( E , E ). To do this, notice that the tensor product ⊗ at right hand sideof (2.6) modifies ⊗ by inserting suitable idempotents p n which may then be disregardedthanks to (2) again. c) Naturality of F and G as transformations from ˜ G → Vec may bechecked with direct computation. Notice also that by construction F and G are naturalas transformations G q → Vec, therefore by composition ˜ G q → G q → Vec we find that theyare also natural with respect to ⊠ . Property d) follows again from (2). (cid:3) Remark 24.11.
It follows from part b) of the previous theorem that the composition Q : ˜ G q → T ( g , q, ℓ ) → C ( g , q, ℓ )of the natural inclusion followed by quotient is an equivalence of tensor categories. Inthis way, ˜ G q admits a unique structure of a ribbon category in a way that Q is a ribbonequivalence. On the other hand, this can also be seen directly. EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 133 A W ( g , q, ℓ ) . In this subsection we assume that q = e iπ/ℓ is a minimal root of large enough order and we study the unitarity property of A W ( g , q, ℓ ).We recall from Theorem 24.1 that U q ( g ) is a (topological) Hermitian coboundary Hopfalgebra with compatible involution and antipode of Kac type. Furthermore recall also thatby Prop. 2.4 in [128], for λ ∈ Λ ℓ the natural Hermitian form of V λ in the sense of Sect. 11is a positive definite inner product, so V λ is a C*-representation of U q ( g ).For completeness———————– Definition 24.12.
Let T W denote the full subcategory of T ( g , q, ℓ ) with objects orthogonaldirect sums of summands defined by selfadjoint idempotents of finite tensor products of V λ with λ ∈ Λ ℓ endowed with the non-degenerate Hermitian form induced by iterates of R of U q ( g ).Consider a finite tensor product W of V λ with λ ∈ Λ ℓ endowed with the non-degenerateHermitian form induced by iterates of R of U q ( g ), or more generally a an orthogonaldirect sum of summands defined by selfadjoint idempotents of a module of this kind suchthat the form is nondegenerate on W . For any morphism T : W → W ′ of T W , theadjoint T ∗ : W ′ → W is well defined. We next consider in particular the canonicaldecomposition into indecomposable tilting modules recalled in the previous subsection V λ ⊗ V = ⊕ γ T γ ⊗ C m γ . The Hermitian form induced by R is positive definite on the onthe isotypic component T γ ⊗ C m γ = V γ ⊗ C m γ for γ ∈ Λ ℓ . The idempotents p λ,γ V λ ⊗ V → V γ ⊗ C m γ are selfadjoint with respect to this inner product. It follows that the iteratedtensor powers V ⊗ n are Hilbert space representations of U q ( g ) with this iterated Hermitianform. Let G ℓ denote the completion under selfadjoint idempotents and orthogonal directsums of the full subcategory of T ( g , q, ℓ ) with objects V ⊗ n . Thus G ℓ has the structure ofa linear semisimple C ∗ -category. Furthermore with tensor product ⊗ defined as in theprevious subsection, G ℓ becomes a unitary strict tensor category. Proposition 24.13.
The idempotents p λ and p λ ⊗ p µ are selfadjoint in G ℓ .Proof. Notice that for all n , the idempotents say p γ,j onto the irreducible decomposition V ( n ) by the V ( n ) γ,j described in Remark 24.3 have pairwise orthogonal ranges with respectto the inner product by orthogonality of the addenda of V λ ⊗ V with dominant weights inΛ ℓ and the iterative construction of V ⊗ n . It follows in particular that p λ are selfadjointidempotent in G ℓ . Let c w ( λ, µ ) be the coboundary operators in T ( g , ℓ, q ) associated to the R matrix of U q ( g ). By naturality we have p λ ⊗ p µ c w ( µ, λ ) = c w ( µ, λ ) p µ ⊗ p λ , and thus R commutes with p λ ⊗ p µ . It follows that p λ ⊗ p µ is selfadjoint with respect to the iteratedHermitian form of V ⊗ n λ + n µ , and therefore also p λ ⊗ p µ are selfadjoint. (cid:3) We then similarly introduce the concrete category ˜ G ℓ of T ( g , q, ℓ ) taking into consider-ation summands defined by selfadjoint idempotents and orthogonal direct sums. By theprevious proposition, for λ , µ ∈ Λ ℓ , F λ,µ and G λ,µ introduced as in the previous subsectionare morphisms in T W and satisfy in addition the property F ∗ λ,µ = G λ,µ . Using orthogonaldecompositions of objects of ˜ G ℓ , we obtain natural transformations F ρ,σ , G ρ,σ as in (24.3),
34 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI and (24.5), satisfying F ∗ ρ,σ = G ρ,σ in T W . Then we introduce in ˜ G ℓ the structure of a ten-sor category (˜ G ℓ , ⊠ , α ) as in (24.4), (24.6), and (24.7). Finally, we consider the forgetfulfunctor, W : ˜ G ℓ → Hilband is a ∗ -functor endowed with the weak tensor structure ( F, G ) regarded with values inHilb. (Notice that as natural transformations of W , we do not have F ∗ ρ,σ = G ρ,σ , moredetails will be discussed in the proof of the following result.) Theorem 24.14.
Let q be a minimal root of large enough order, then a) (˜ G ℓ , ⊠ , α ) is a unitary semisimple tensor category and the tensor equivalence ( E , E ) :(˜ G ℓ , ⊠ , α ) → ( G ℓ , ⊗ , is unitary, b) A W ( g , q, ℓ ) = Nat ( W ) becomes a unitary coboundary weak w-Hopf algebra withcompatible involution, weak tensor structure defined by ( F, G ) and antipode of Kactype such that G ℓ → Rep( A W ( g , q, ℓ )) is a unitary equivalence of ribbon categories.Proof. a) The property ( S ⊠ T ) ∗ = S ∗ ⊠ T ∗ follows from the relation F ∗ = G in T Λ ℓ andarguments similar to those in the proof of Prop. 24.13. Unitarity of the associator followsfrom F ∗ = G and (2), and (3). b) By theorem Theorem 22.1 we need to show (22.1) and(22.2). We only show the former. By Remark 22.2 a), it is enough to do this for ρ = V ⊗ n , σ = V . In this case F ρ,σ = p n +1 as (1 − p n ) ⊗ V is negligible. This follows by constructionas F ∗ ρ,σ = p ∗ n +1 R = R where p ∗ n +1 is the adjoint with respect to the standard inner productof V ⊗ n ⊗ V . We next show that A W ( g , q, ℓ ) has an antipode of Kac type. It is shown inthe proof of Lemma 10.4 in [23] that a solution of the conjugate equations in G ℓ is of theform (11.1) with α = β = I , µ i = 1. It follows from the proof of Theorem 5.6 that thecorresponding antipode is strong and therefore of Kac type. The proof of compatibilitywith the ∗ -involution is the content of the following lemmas 24.15, 24.16, 24.17. (cid:3) Lemma 24.15.
Let c U ( ρ, σ ) be the natural coboundary symmetry associated to U q ( g ) . Thenthe unitary coboundary w-Hopf algebra A W has compatible ∗ -involution if and only if c U ( V λ , V ⊗ k ) G V λ ,V ⊗ k F V λ ,V ⊗ k c U ( V ⊗ k , V λ ) = G V ⊗ k ,V λ F V ⊗ k ,V λ λ ∈ Λ ℓ , k = 1 , . (24.8) It suffices that the following two equations involving the braided symmetries c U and c − U associated to U q ( g ) and also the braided symmetries c and c − associated to A W , hold, c ( V λ , V ⊗ k ) F V λ ,V ⊗ k c U ( V λ , V ⊗ k ) − = F V ⊗ k ,V λ , λ ∈ Λ ℓ , k = 1 , , (24.9) c ( V ⊗ k , V λ ) − F V λ ,V ⊗ k c U ( V ⊗ k , V λ ) = F V ⊗ k ,V λ , λ ∈ Λ ℓ , k = 1 , . (24.10) Proof.
Taking the adjoint of equations (24.9), (24.10) and multiplying them term by termwe get c U ( V λ , V ⊗ k ) G V λ ,V ⊗ k F V λ ,V ⊗ k c U ( V λ , V ⊗ k ) − = G V ⊗ k ,V λ F V ⊗ k ,V λ λ ∈ Λ ℓ , k = 1 , . (24.11) EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 135 c U ( V ⊗ k , V λ ) − G V λ ,V ⊗ k F V λ ,V ⊗ k c U ( V ⊗ k , V λ ) = G V ⊗ k ,V λ F V ⊗ k ,V λ λ ∈ Λ ℓ , k = 1 , . (24.12)In turn it follows that c U commutes with G V ⊗ k ,V λ F V ⊗ k ,V λ . It follows that the principalbranch square root commutes also, and this implies (24.8).We next show the first statement. By Prop. 23.2, compatibility of the ∗ -involution isequivalent to ∆ op ( I ) = ∆( I ) ∗ on the spaces of V ⊗ k ⊗ V λ and V λ ⊗ V ⊗ k for k = 1, 2.We have ∆( I ) ∗ = R U ∆( I )( R U ) − , and it follows that the desired equalities reduce to ourassumptions. Note that equation (24.8) together with the coboundary property c U = 1imply that the symmetric equation with V λ on the right and V ⊗ k on the left at the l.h.s.of the equation holds and this completes the proof. (cid:3) Lemma 24.16.
The natural transformation F defining A W satisfies equations (24.9),(24.10) for k = 1 .Proof. Assume g = E . For k = 1, by [128], V ⊗ V λ is completely reducible into irreduciblecomponents ⊕ µ m µ V µ (with multiplicity 0 or 1 except for g = F where µ µ > µ = λ ) and we have that µ ∈ Λ ℓ . Thus there is a unique morphism idempotent onto amaximal non-negligible submodule V ⊗ V λ → ⊕ µ ∈ Λ ℓ V µ which then coincides with F V,V λ .This uniqueness property and unitarity of the braided symmetries imply that (24.9), (24.10)hold for k = 1. The case g = E is more delicate that the others, and is not covered bythe above proof. In this case we consider the decomposition of V λ ⊗ V and of V ⊗ V λ intoindecomposable tilting modules given at page 274 in [128]. Let F V λ ,V and F V,V λ be thecorresponding idempotents onto the maximal non-negligible submodules. Then it followsfrom the proof therein and unitarity of the braided symmetry, that (24.9), (24.10) hold inthis case. (cid:3) Lemma 24.17.
The natural transformation F defining A W satisfies equations (24.9),(24.10) for k = 2 and all Lie types.Proof. We use the w-Hopf property in categorical form (2.6), (2.7). Working with Vecstrict, F ( α V λ ,V,V ) = F V λ ,V ⊗ V ◦ ⊗ F V,V ◦ G V λ ,V ⊗ ◦ G V λ ⊗ V,V , (24.13) F ( α V λ ,V,V ) − = F V λ ⊗ V,V ◦ F V λ ,V ⊗ ◦ ⊗ G V,V ◦ G V λ ,V ⊗ V , (24.14) F ( α V,V,V λ ) = F V,V ⊗ V λ ◦ ⊗ F V,V λ ◦ G V,V ⊗ ◦ G V ⊗ V,V λ , (24.15) F (( α V,V,V λ ) − ) = F V ⊗ V,V λ ◦ F V,V ⊗ ◦ ⊗ G V,V λ ◦ G V,V ⊗ V λ . (24.16)We set ˜ F , := F V λ ,V ⊗ V ◦ ⊗ F V,V ,G , = G V λ ,V ⊗ ◦ G V λ ⊗ V,V ,
36 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI F , = F V λ ⊗ V,V ◦ F V λ ,V ⊗ , ˜ G , = 1 ⊗ G V,V ◦ G V λ ,V ⊗ V .F ′ , = F V,V ⊗ V λ ◦ ⊗ F V,V λ , ˜ G ′ , = G V,V ⊗ ◦ G V ⊗ V,V λ , ˜ F ′ , = F V ⊗ V,V λ ◦ F V,V ⊗ ,G ′ , = 1 ⊗ G V,V λ ◦ G V,V ⊗ V λ . Note that by Lemma 24.16, naturality of all the transformations and the braiding, andthe two hexagonal equations (3.8), (3.9), the map G , , ( F , resp.), is conjugate to G ′ , ,( F ′ , resp.) via a specific braiding (that is the representative of the braid group element b b b = b b b in the category) For example, F V,V ⊗ V λ = c ( V ⊗ V λ , V ) F V ⊗ V λ ,V c U ( V ⊗ V λ , V ) − ⊗ F V,V λ = 1 ⊗ c ( V λ , V ) ◦ ⊗ F V λ ,V ◦ ⊗ c U ( V λ , V ) − imply F ′ , = c ( V ⊗ V λ , V ) c ( V λ , V ) ⊗ ◦ F , ◦ (1 ⊗ c U ( V λ , V ) c U ( V ⊗ V λ , V )) − = c ( V ⊗ V λ , V ) c ( V λ , V ) ⊗ ◦ F , ◦ ( c U ( V ⊗ V λ , V ) c U ( V λ , V ) ⊗ − . Multiplying together (24.13) and (24.14) and then (24.15) and (24.16) gives respectively1 = F , ◦ ⊗ G V,V ◦ P V λ ,V ⊗ V ◦ ⊗ F V,V ◦ G , , (24.17)1 = F ′ , ◦ G V,V ⊗ ◦ P V ⊗ V,V λ ◦ F V,V ⊗ ◦ G ′ , . (24.18)Conjugating (24.18) by the same braid group element gives1 = F , ◦ ⊗ G V,V ◦ P cV ⊗ V,V λ ◦ ⊗ F V,V ◦ G , , (24.19)where P V λ ,V ⊗ V = G V λ ,V ⊗ V F V λ ,V ⊗ V ,P cV ⊗ V,V λ = c U ( V ⊗ V, V λ ) ◦ G V ⊗ V,V λ ◦ F V ⊗ V,V λ ◦ c U ( V ⊗ V, V λ ) − . It follows from (24.17) and (24.19) that0 = F , ◦ ⊗ G V,V ◦ A ◦ ⊗ F V,V ◦ G , , (24.20)where A = P V λ ,V ⊗ V − P cV ⊗ V,V λ may be regarded a selfadjoint element of a C ∗ -algebra, henceit can be written as the difference of two orthognal positive operators A = A + − A − , A + A − = A − A + = 0 . Being G , F , an idempotent onto a maximal non-negligible submodule, we have0 = F , ◦ ⊗ G V,V ◦ A + ◦ ⊗ F V,V ◦ G , F , ◦ ⊗ G V,V ◦ A − ◦ ⊗ F V,V ◦ G , . It follows from (24.20) that0 = F , ◦ ⊗ G V,V ◦ A ◦ ⊗ F V,V ◦ G , , (24.21)0 = F , ◦ ⊗ G V,V ◦ A − ◦ ⊗ F V,V ◦ G , (24.22) EAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CFT 137
Taking the categorical trace we have Tr( A ) = Tr( A − ) = 0 , hence A + = A − = 0 by the C ∗ -property and it follows that A = 0, that is P V λ ,V ⊗ V = P cV ⊗ V,V λ . (cid:3) Compatible unitary coboundary wqh algebra structure on the Zhualgebra A ( V g k ) as a subquotient of U ( g ) , connection with work byCKLW, CWX, FKL equivalence theorem In this section we are interested in the affine vertex operator algebras. This is animportant class of vertex operator algebras associated to affine Lie algebras at positiveinteger levels. Every vertex operator algebra has an associated associative algebra, calledthe Zhu algebra [137] briefly recalled in Sect. 17. We refer to [43] for the definition ofVertex Operator Algebra and the associated Zhu algebra. We shall briefly recall a naturalidentification of the Zhu algebra in the case of affine VOAs. We postpone a more completepresentation to an updated version of the paper.
Affine Lie algebra ˆ g , VOA V g k , Zhu algebra A ( V g k ) . Let g be a complex finite dimensionalsimple Lie algebra, h a Cartan subalgebra, α , . . . , α r a set of simple roots, and A = ( a ij )the associated Cartan matrix. Consider the unique invariant symmetric and bilinear formon h ∗ such that hh θ, θ ii = 2 where θ denotes the highest root. Consider the affine Liealgebra ˆ g = g ⊗ C [ t, t − ] ⊕ C k , with k in the center of ˆ g and Lie algebra structure given by[ a ⊗ t n , b ⊗ t m ] = [ a, b ] ⊗ t m + n + k hh a, b ii δ m + n, . Let us fix k ∈ C . Every g -module W gives rise to a ˆ g -module W k such that k actsas the scalar k . For a fixed irreducible g -module L ( λ ) with dominant weight λ ∈ h ∗ ,corresponding ˆ g -module L k,λ is characterized up to isomorphism by the following threeproperties, i) L k,λ is irreducible, ii) k acts as k , iii) L k,λ contains an isomorphic copy of L ( λ ) given by { a ∈ L k,λ , ˆ g + a = 0 } , where ˆ g + = g ⊗ C [ t ] t . By [43], V g k := L k, has thestructure of a vertex operator algebra for k = h ∨ , the dual Coxeter number and when k isa positive integer, V g k is a rational VOA, see also Sect. 17 for more details and referencesto the original papers. By Theorem 3.1.2 in [43], in this case the Zhu algebra A ( V g k ) iscanonically isomorphic to a quotient of U ( g ) (by the two-sided ideal generated by e k +1 θ ,where e θ is an element in the root space g θ of the maximal root θ .) By Theorem 3.1.3in [43], the set L λ,k , where λ is a dominant weight with hh λ, θ ii ≤ k is a complete list ofirreducible V g k -modules.A detailed proof the the following theorem will be given in an update of this version. Theorem 25.1.
The Zhu algebra A ( V g k ) admits a canonical structure of compatible unitarycoboundary weak quasi-Hopf algebra with strongly unitary structure obtained by transferringthe untwisted structure of A W ( g , q, ℓ ) via Drinfeld-Kohno theorem and Wenzl continuouspath argument. The ∗ -involution and unitary structure on V g k -modules coincides with thatof [20] , [21] . We may then apply Tannakian theorems.
38 S. CARPI, S. CIAMPRONE, M.V. GIANNONE, AND C. PINZARI
Corollary 25.2.
The linear category
Rep( V g k ) becomes a unitary modular tensor categorywith the structure introduced in the previous theorem. Final conjectures
We conjecture that the modular tensor category structure introducedin the previous corollary coincides with that given by Huang and Lepowsky. This conjectureimplies a new proof of Kazhdan-Lusztig-Finkelberg theorem based on unitarity. We alsoconjecture that the hermitian form of the unitary wqh A ( V g k ) coincides with that of Gui.We shall study these conjectures and their implications in future work. Acknowledgments.
We would like to thank J. Barrett, A. D’Andrea, S. Del Vecchio, L.Giorgetti, B. Gui, A. Henriques, S. Iovieno, S. Neshveyev, D. Reutter, L. Tuset, H. Wenzl,M. Yamashita, P. Zurlo for interest in our work and fruitful discussions.
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