aa r X i v : . [ m a t h . QA ] M a y GLUEING VERTEX ALGEBRAS
THOMAS CREUTZIG, SHASHANK KANADE AND ROBERT MCRAE
Abstract.
Let U and V be vertex operator algebras with module (sub)categories U and V , respectively, satisfying suitable assumptions which hold for example if U and V aresemisimple rigid braided (vertex) tensor categories with countably many inequivalent simpleobjects. If τ is a map from the set of inequivalent simple objects of U to the objects V with τ ( U ) = V , then we glue U and V along U ⊠ V via τ to obtain the object A = M X ⊗ τ ( X )where the sum is over all inequivalent simple objects of U . Assuming U and V form acommuting pair in A in the sense that the multiplicity of V is U , our main theorem is thatthere is a braid-reversed equivalence between U and V mapping X to τ ( X ) ∗ if and only if A can be given the structure of a simple conformal vertex algebra that (conformally) extends U ⊗ V . Contents
1. Introduction 21.1. Tensor category results 21.2. Applications to vertex operator algebras 41.3. Examples 61.4. Outlook 8Acknowledgements 82. Algebras in braided tensor categories 92.1. Braided tensor categories 92.2. Direct sum completion 102.3. Representation categories of an algebra object 122.4. The center of a tensor category 173. From braid-reversed equivalences to algebras 203.1. Associative algebras 203.2. Commutative algebras 233.3. Canonical algebras 254. From algebras to braid-reversed equivalences 284.1. Mirror equivalence 284.2. Proof of Key Lemma 4.2 and Proposition 4.4 325. From tensor categories to vertex operator algebras 435.1. Vertex tensor categories 435.2. Deligne products of vertex algebraic tensor categories 455.3. Algebras in vertex tensor categories 495.4. The main theorems for vertex operator algebras 51Appendix A. Direct sum completion 53 eferences 551. Introduction
We study the relation between certain types of commutative associative algebra objects inbraided tensor categories and braid-reversed equivalences of tensor categories, motivated byvertex operator algebra theory and its applications to geometry and physics. Commutativeassociative algebra objects in tensor categories of modules for a vertex operator algebra arethe same as vertex operator algebra extensions [HKL], and such extensions together withequivalences of vertex tensor subcategories are crucial in the context of S -duality for four-dimensional supersymmetric GL -twisted gauge theories [CGai] and the quantum geometricLanglands correspondence [AFO]. In gauge theory, vertex operator algebras are associatedto two-dimensional intersections of three-dimensional topological boundary conditions, whilecategories of vertex operator algebra modules are associated to line defects ending on theseboundary conditions. Boundary conditions can be concanated to form new types of boundaryconditions, and the resulting vertex operator algebras are precisely the type of extensionsstudied in this work. Categories of vertex operator algebra modules appearing in theseproblems are usually not finite and are often, but not necessarily, semisimple. Thus wederive results in a setting general enough for these applications, especially allowing braidedtensor categories to have infinitely many inequivalent simple objects.We will now describe our categorical results, followed by vertex operator algebra applica-tions and comments regarding vertex operator algebra theory and existing literature.1.1. Tensor category results.
Let C and D be braided tensor categories and τ a map fromsimple objects in C to objects in D . Then we consider objects in a direct sum completion( C ⊠ D ) ⊕ of the Deligne product C ⊠ D of the form A = M X ∈ Irr( C ) X ⊠ τ ( X ) ∈ Obj (( C ⊠ D ) ⊕ ) . We aim to prove under suitable conditions on C and D that a commutative associativealgebra structure on A is equivalent to a braid-reversed tensor equivalence between C and D .Thus we ask two questions: under which conditions on C and D does a commutative algebraobject imply a braid-reversed equivalence, and conversely what do we need to assume sothat a braid-reversed equivalence yields a commutative associative algebra?1.1.1. From braid-reversed equivalences to commutative algebra objects.
It is known (see for example [EGNO, Exercise 7.9.9]) that associative algebras can beconstructed from module categories; as we could not find a complete proof in the literature,we give this construction in Theorem 3.2. Essentially, for C a (multi)tensor category and M a C -module category such that internal Homs exist, the internal End of an object M in M can be given the structure of an associative algebra. Moreover, when C is a braidedfusion category, [DMNO, Lemma 3.5] shows that this object is even commutative as longas induction from C to M is a central functor, that is, it factors through the center ofthe module category. In Theorem 3.3, we prove this statement in detail without assumingfiniteness or semisimplicity. useful property in showing that internal Homs exist is rigidity, that is, existence ofduals. Unfortunately, braided tensor categories of vertex operator algebra modules are oftennot known to be rigid. However, we find that internal Homs can exist under the weakerassumption that C has a contragredient functor, that is, a contravariant endofunctor X X ′ such that there is a natural isomorphismHom C ( X ⊗ Y , ) ∼ = Hom C ( X , Y ′ )for objects X , Y in C . For vertex algebraic tensor categories, such a functor arises fromthe contragredient modules of [FHL], provided that the vertex operator algebra itself isself-contragredient.With these preparations, we can state our first main result; for precise notation we referto Section 3.3. The algebra of this theorem is called the canonical algebra in C ⊠ C rev , where C rev = C as a tensor category but has reversed braidings. Main Theorem 1.
Let C be a (not necessarily finite) semisimple braided tensor categorywith a contragredient functor. Then A = M X ∈ Irr( C ) X ′ ⊠ X is a commutative associative algebra in ( C ⊠ C rev ) ⊕ . If C is rigid, then A is simple and forsimple objects X , Y , Z of C , the multiplication rules are given by M Z ∗ ⊠ ZX ∗ ⊠ X , Y ∗ ⊠ Y ∗ = 1 if and onlyif Z is a summand of X ⊗ Y . Since commutative algebras are preserved by braided tensor equivalences, we can restateMain Theorem 1 as follows. Let C be a semisimple braided tensor category with a contra-gredient functor, and suppose τ : C → D is a braid-reversed tensor equivalence (so that τ : C rev → D is a braided equivalence). Then A = M X ∈ Irr( C ) X ′ ⊠ τ ( X )is a commutative associative algebra in ( C ⊠ D ) ⊕ , and if C is rigid, then A is simple.1.1.2. From commutative algebra objects to braid-reversed equivalences.
For the converse question, we work in the following setting:(1) U is a (not necessarily finite) semisimple ribbon category, and { U i } i ∈ I is a subset ofdistinct simple objects in U that includes U = U .(2) V is a ribbon category. In particular, both U and V are rigid.(3) We have a simple (commutative, associative, unital) algebra A = M i ∈ I U i ⊠ V i . in C = U ⊠ V , or C ⊕ if I is infinite, where the V i are objects of V , not assumed to besimple except for V = V .(4) The tensor units U = U , V = V form a mutually commuting (or dual) pair in A ,in the sense thatdim Hom U ( U , U i ) = δ i, = dim Hom V ( V , V i ) .
5) There is a partition I = I ⊔ I of the index set such that 0 ∈ I and for each i ∈ I j , j = 0 ,
1, the twist satisfies θ A | U i ⊠ V i = ( − j Id U i ⊠ V i . In particular, θ A = Id A .Under these conditions, we define U A ⊆ U and V A ⊆ V to be the full subcategories whoseobjects are isomorphic to direct sums of the U i and V i , respectively, and prove in Proposition4.4 and Theorem 4.5: Main Theorem 2.
In the setting of this section, (1)
The categories U A ⊆ U and V A ⊆ V are ribbon subcategories. Moreover, V A issemisimple with distinct simple objects { V i } i ∈ I . (2) There is a braid-reversed tensor equivalence τ : U A → V A such that τ ( U i ) ∼ = V ∗ i for i ∈ I . This theorem relies on the following Key Lemma; here F is the the induction functor from C to the category Rep A of left A -modules in C : Key Lemma 1.
For all i ∈ I , F ( U i ⊠ V ) ∼ = F (( U ⊠ V i ) ∗ ) in Rep A . Applications to vertex operator algebras.
Our categorical results translate intothe following theorem for vertex operator algebras; see Theorem 5.10 of the main text:
Main Theorem 3.
Let U and V be locally finite module categories for simple and self-contragredient vertex operator algebras U and V , respectively, that are closed under con-tragredients and admit vertex tensor category structure as in [HLZ1] - [HLZ8] and thus alsobraided tensor category structure. Assume moreover that U is semisimple and V is closedunder submodules and quotients. (1) Suppose { U i } i ∈ I is a set of representatives of equivalence classes of simple modulesin U with U = U and τ : U → V is a braid-reversed tensor equivalence with twistssatisfying θ τ ( U i ) = ± τ ( θ − U i ) for i ∈ I . Then A = M i ∈ I U ′ i ⊗ τ ( U i ) is a Z -graded conformal vertex algebra extension of U ⊗ V . Moreover, if U is rigid,then A is simple and the multiplication rules of A satisfy M U ′ k ⊗ τ ( U k ) U ′ i ⊗ τ ( U i ) , U ′ j ⊗ τ ( U j ) = 1 if andonly if U k occurs as a submodule of U i ⊠ U j . (2) Conversely, suppose U and V are both ribbon categories, { U i } i ∈ I is a set of distinctsimple modules in U with U = U , and A = M i ∈ I U i ⊗ V i is a simple Z -graded conformal vertex algebra extension of U ⊗ V , where the V i areobjects of V satisfying dim Hom V ( V , V i ) = δ i, and there is a partition I = I ⊔ I of the index set with ∈ I and M i ∈ I j U i ⊗ V i = M n ∈ j + Z A ( n ) for j = 0 , . Let U A ⊆ U , respectively V A ⊆ V , be the full subcategories whose objectsare isomorphic to direct sums of the U i , respectively of the V i . Then: a) U A and V A are ribbon subcategories of U and V respectively. Moreover, V A issemisimple with distinct simple objects { V i } i ∈ I . (b) There is a braid-reversed equivalence τ : U A → V A such that τ ( U i ) ∼ = V ′ i for all i ∈ I . Conformal vertex algebra extensions as in the first part of the theorem have previouslybeen constructed for certain affine Lie algebra [FS, Zhu] and Virasoro [FZ] vertex operatoralgebras. Also, a closely-related construction due to Huang and Kong [HK, Ko], starting frombraid-equivalent modular tensor categories of representations for vertex operator algebras,yields a conformal full field algebra in the sense of [HK]. In fact, [Ko] shows that if U and V are braid-equivalent tensor categories of representations for strongly rational vertex operatoralgebras U and V , respectively, then conformal full field algebra extensions of U ⊗ V withnondegenerate invariant bilinear form are equivalent to commutative Frobenius algebras in U ⊠ V rev with trivial twist.The second part of Main Theorem 3 (in the case that A is Z -graded) has been stated in[Lin, Theorem 3.3] under the assumption that U and V are strongly rational vertex operatoralgebras (in particular I is finite in this setting). The proof in [Lin] uses semisimplicity of thecategory Rep A of left A -modules in C , citing [KO] for this result. However, [KO, Theorem 3.3]assumes additionally that dim C A = 0 to prove this semisimplicity, whereas even a modulartensor category can have objects with dimension zero. Relaxing the condition dim C A = 0is the work of our Key Lemma 1, so we have in particular filled a gap in [Lin]; moreover, werecover the semisimplicity of the category of A -modules as a consequence, as we now discuss.A vertex operator algebra is strongly rational (in the sense of [CGan]) if it is simple,self-contragredient, CFT-type, C -cofinite, and rational; for such a vertex operator algebra,the full category of grading-restricted, generalized modules is a (semisimple) modular tensorcategory [Hu3]. Let V be a strongly rational vertex operator algebra and A a simple CFT-typevertex operator algebra extension of V ; then A is believed to have a modular tensor categoryof grading-restricted, generalized modules as well. This indeed follows from Lemma 1.20,Theorem 3.3, and Theorem 4.5 of [KO] as well as [HKL, Theorem 3.5] (see also [DMNO,Corollary 3.30]) provided the dimension of A as a V -module is non-zero. Moreover, ourprevious work [CKM, Theorem 3.65] shows that this modular tensor category structure isthe natural one for module categories of a vertex operator algebra.Using [Hu3, DJX], the dimension of A in the modular tensor category C of V -modules isstrictly positive if all irreducible V -modules are non-negatively graded, with a non-zero con-formal weight 0 space occurring only in V itself. This condition together with the rationalityand C -cofiniteness of V ensures that the categorical dimensions of V -modules are realized bystrictly-positive “quantum dimensions” defined in terms of characters. Now, we can use thebraid-reversed equivalence of Main Theorem 3 and [ENO, Theorem 2.3] to calculate dim C A without grading-positivity assumptions and conclude (see Corollary 5.13 in the main text): Corollary 1.1.
In the setting of Main Theorem 3, assume in addition that U and V arestrongly rational and A is a simple CFT-type ( Z -graded) vertex operator algebra. Then dim C A > and A is strongly rational; in particular its category of grading-restricted, gener-alized modules is a semisimple modular tensor category. In fact, the calculation of dim C A does not require rationality or C -cofiniteness for either U or V , so we prove more generally that if the categories U and V in Main Theorem 3 are raided fusion categories, then the category of grading-restricted generalized A -modules in C is also braided fusion; see Theorem 5.12 in the main text for details. In our setting, thisremoves the grading-positivity assumptions from [HKL, Theorem 3.5].A second corollary of Main Theorem 3 relates to the multiplication rules of the extension U ⊗ V ⊆ A . In general, if A = L i ∈ I V i is an extension of a vertex operator algebra V = V byindecomposable V -modules V i , define the multiplication rule M ki,j to be 1 if V k is containedin the operator product algebra of fields of V i with fields of V j ; otherwise the multiplicationrule is 0. It is clear that M ki,j = 0 if the fusion rule N ki,j = 0; a question raised in privatecommunication by Chongying Dong is for which extensions is the converse also true. Inour setting, this is precisely the content of the statements on multiplication rules in MainTheorems 1 and 3. Thus we can rephrase these statements as follows: Corollary 1.2.
In the setting of part (1) of Main Theorem 3, and assuming U is rigid, themultiplication rules for the canonical algebra A = M i ∈ I U ∗ i ⊗ τ ( U i ) are if and only if the corresponding fusion rules are . Examples.
We illustrate our results in two examples; the first illustrates both partsof Main Theorem 3 and shows the importance of allowing categories with infinitely manysimple objects.1.3.1.
Vertex algebras for S -duality and Kazhdan-Lusztig categories at generic level. Thefollowing conjecture is the physics prediction of vertex operator algebras associated to theintersection of so-called Dirichlet boundary conditions and their general S -duals. Thesevertex operator algebras are claimed in [CGai] to play a role as quantum geometric Langlandskernel vertex operator algebras. Conjecture 1.3. [CGai, Conjecture 1.1]
Let g be a simple Lie algebra and P + n the set of alldominant weights λ such that nλ is integral. Let ψ, ψ ′ be generic complex numbers satisfying ψ + 1 ψ ′ = n. Then the object A n [ g , ψ ] := M λ ∈ P + n L ψ − h ∨ ( λ ) ⊗ L ψ ′ − h ∨ ( λ ) can be given the structure of a simple vertex operator superalgebra. This conjecture is known to be true for n = 0 as we will explain in a moment and also for g = sl and n = 1 [CGai] and n = 2 [CGL].Let κ be an irrational number. The algebra of chiral differential operators of a compactLie group G [GMS1, GMS2, AG] with Lie algebra g at level κ has the form D ch κ ( G ) ∼ = M λ ∈ P + L κ − h ∨ ( λ ) ⊗ L − κ − h ∨ ( λ ∗ )as a L κ − h ∨ ( g ) ⊗ L − κ − h ∨ ( g )-module and is a simple vertex operator algebra ([FS, Zhu] and[Ch, Proposition 3.15]). Here L k ( λ ) denotes the irreducible highest-weight module of highest-weight λ at level k ; also, λ ∗ = − ω ( λ ), where ω is the longest element of the Weyl group. By KL1]-[KL4] and [Zha] (see [Hu4] for a review), the category of such highest-weight modulesat irrational level κ , denoted by KL κ is a semisimple rigid vertex tensor category. It followsthat D ch κ ( G ) is a commutative algebra object in the Deligne product KL κ − h ∨ ⊠ KL − κ − h ∨ ,and moreover the categories KL κ − h ∨ and KL − κ − h ∨ are braid-reversed equivalent by part (2)of Main Theorem 3 with objects L κ − h ∨ ( λ ) identified with L − κ − h ∨ ( λ ) under the equivalence.On the other hand, KL κ − h ∨ is also equivalent as a braided tensor category to the category ofweight modules U q ( g )-mod (representations of type I) of the Lusztig quantum group U q ( g )for q = exp (cid:0) πir ∨ κ (cid:1) [KL1]-[KL4] for r ∨ the lacity of g .Let N be the level of the weight lattice P of g , that is the smallest positive integer suchthat N P is integral. The representation category of the rational form of the quantum groupis over Q ( s ) with s = exp (cid:0) πiNr ∨ κ (cid:1) ([Lu]; see also [BK, Section 1.3]) and so we have thatKL κ − h ∨ and KL ℓ − h ∨ are equivalent if κ = ℓ mod r ∨ N . Combining with the braid-reversedequivalence we have that KL κ − h ∨ and KL ℓ − h ∨ are braid-reversed equivalent if1 κ + 1 ℓ = mr ∨ N for some m ∈ Z , so that by part (1) of Main Theorem 3 M λ ∈ P + L κ − h ∨ ( λ ) ⊗ L ℓ − h ∨ ( λ ∗ )for such κ and ℓ has the structure of a simple vertex operator algebra. We can change thesimple root system for the second factor by − ω so that A mr ∨ N [ g , κ ] = M λ ∈ P + L κ − h ∨ ( λ ) ⊗ L − κ − h ∨ ( λ )also can be given the structure of a simple vertex operator algebra. This proves Conjecture1.1 of [CGai] for n = mr ∨ N , that is, Corollary 1.4.
Let N be the level of the weight lattice P of the simple Lie algebra g . Then [CGai, Conjecture 1.1] is true for n ∈ N r ∨ Z . Equivalences between affine vertex operator algebras and W -algebras at admissiblelevel. There are also interesting equivalences of vertex tensor categories at admissible levels:let g be a simple simply-laced Lie algebra, h ∨ the dual Coxeter number, and k an admissiblelevel of g . Let P m + be the set of weights λ such that the irreducible highest-weight representa-tion L m ( λ ) is a module of the simple affine vertex operator algebra L m ( g ) at positive integerlevel m . We parameterize k = − h ∨ + uv so that the simple ordinary modules of the simpleaffine vertex operator algebra L k ( g ) of g at level k are the irreducible highest-weight modules L k ( λ ) of highest-weight λ and level k with λ ∈ P u − h ∨ + . These modules are called ordinaryand the category of ordinary modules at admissible level for simply-laced g is semisimple[Ar2], vertex tensor [CHY], and rigid [Cr]. Let W k ( g ) be the simple principal W -algebra of g at level k . It is strongly rational if k is non-degenerate admissible [Ar1], that is, u, v ≥ h ∨ inthe simply-laced case. We denote the image of L k ( λ ) under quantum Hamiltonian reductionby W k ( λ ). Let ℓ satisfy 1 k + 1 + h ∨ + 1 ℓ + h ∨ = 1 . ith this notation a special case of [ACL, Main Theorem 3 (1)] says that L k ( g ) ⊗ L ( g ) ∼ = M λ ∈ P u + v − h ∨ + ∩ Q L k +1 ( λ ) ⊗ W ℓ ( λ )as L k +1 ( g ) ⊗ W ℓ ( g )-modules. Let us denote the subcategory of ordinary modules whoseweights lie in the root lattice Q by O Qk, ord ( g ) and their image under quantum Hamiltonianreduction by C Qk, ord ( W ( g )). Applying our Main Theorem 3 we have Corollary 1.5.
Let k be admissible and g simply-laced, and define ℓ by k + 1 + h ∨ + 1 ℓ + h ∨ = 1 . Then there is a braid-reversed equivalence between O Qk +1 , ord ( g ) and C Qℓ, ord ( W ( g )) sending L k +1 ( λ ) to W ℓ ( λ ) ∗ . Outlook.
Our research program aims to understand representation categories of vertexoperator algebras using techniques of the theory of tensor categories. In this paper, we haveproven that vertex operator algebra extensions of suitable tensor products of two vertexoperator algebras are possible if and only if certain subcategories of modules of the twovertex operator algebras are braid-reversed equivalent tensor categories. These results fallinto the area of coset vertex algebras, as the commutant of a vertex subalgebra V ⊆ A is called the coset C of V in A . Often one is dealing with problems where one knows A and V fairly well and would like to study the coset vertex operator algebra C . The veryfirst statement we then need is the existence of vertex tensor category structure on suitablecategories of C -modules. Such an existence result has recently been obtained in the relatedcontext of orbifold vertex operator algebras [McR], and we hope to extend these results tothe more complicated setting of cosets.Ultimately, one of the deepest problems in the area is the conjecture that the coset vertexoperator algebra of a strongly rational vertex operator algebra A by a strongly rationalvertex subalgebra V is itself strongly rational. Theorem 7.6 of [FFRS] is a good guide as itgives a relation between the category of modules for the coset vertex operator algebra andthe categories of A - and V -modules without assuming semisimplicity of the coset modulecategory. It is however still proven under very strong assumptions, such as separability(traces of idempotents are non-zero). We now have techniques to prove statements avoidingassumptions like separability, and we hope to use them to come closer to the rationalityconjecture for coset vertex operator algebras.A second application is that interesting affine vertex operator superalgebras can be re-alized as extensions of affine vertex operator algebras and W -algebras, and then using ourtheory in [CKM] one can study the representation theory of the superalgebras. This hasfor example been succesfully applied to L k ( osp (1 | L k ( sl ) times rational Virasoro algebras [CFK, CKLiuR]. Thus another further goal is toextend the results of this work to superalgebras and construct many more interesting vertexsuperalgebras. With this in mind, we have already proved Key Lemma 4.2 and Proposition4.4 for (supercommutative) superalgebras. Acknowledgements.
TC thanks Fedor Malikov for discussions on the algebra of chiraldifferential operators, and we all thank Yi-Zhi Huang for discussions and comments. TC s supported by NSERC Algebras in braided tensor categories
In this section, we review basic definitions and properties of (braided) tensor categoriesand (commutative, associative) algebra objects.2.1.
Braided tensor categories.
Here we recall some definitions and structures in tensorcategories.
Definition 2.1.
Let C be a category with a distinguished object C and bifunctor ⊗ : C ×C →C . We say that C is a tensor category if(1) For any object X , there are natural isomorphisms l X : C ⊗ X ∼ = −→ X (left unit isomor-phism) and r X : X ⊗ C ∼ = −→ X (right unit isomorphism),(2) For any triple of objects X , Y , Z , there is a natural associativity isomorphism A X , Y , Z : X ⊗ ( Y ⊗ Z ) ∼ = −→ ( X ⊗ Y ) ⊗ Z ,(3) The isomorphisms l , r , A satisfy the triangle axiom and the associativity A satisfiesthe pentagon axiom.A tensor category C is a braided tensor category if additionally:(1) For all pairs of objects X , Y , there is a natural braiding isomorphism R X , Y : X ⊗ Y → Y ⊗ X (2) The isomorphisms R satisfy the hexagon axioms.A tensor category C is rigid if every object has a left and a right dual. We shall only neednotation for the left dual: for any object X we denote the evaluation map by e X : X ∗ ⊗ X → and the coevaluation map by i X : → X ⊗ X ∗ .A rigid braided tensor category C is ribbon if there is a natural isomorphism θ : Id C → Id C ,called the twist , satisfying:(1) θ C = Id C ,(2) θ X ∗ = ( θ X ) ∗ , and(3) The balancing axiom : θ X ⊗ Y = R Y , X ◦ R X , Y ◦ ( θ X ⊗ θ Y ).We will sometimes need to consider tensor categories that are not or are not known tobe rigid, since they may lack coevaluations. In these cases, however, some consequences ofrigidity still hold when C has a weaker structure: we say that a contravariant functor C → C ,which we shall denote by X X ′ , f f ′ , is a contragredient functor if it permutes thesimple objects of C and there are natural isomorphismsΓ X , Y : Hom C ( X ⊗ Y , ) → Hom C ( X , Y ′ ) , natural in the sense that for morphisms f : X → X and g : Y → Y in C , the diagramHom C ( X ⊗ Y , ) Γ X , Y / / F F ◦ ( f ⊗ g ) (cid:15) (cid:15) Hom C ( X , Y ′ ) G g ′ ◦ G ◦ f (cid:15) (cid:15) Hom C ( X ⊗ Y , ) Γ X , Y / / Hom C ( X , Y ′ ) (2.1) ommutes. Given a contragredient functor, we shall denote the morphism Γ − X ′ ⊗ X (Id X ′ ) : X ′ ⊗ X → by e X and call it the evaluation for X . Note that if C is rigid and braided, theduality functor ∗ is a contragredient functor with the natural isomorphisms Γ X , Y obtainedusing i Y . An example of a non-rigid category with a contragredient functor is the categoryof all vector spaces over a field. Remark 2.2. If C is braided, we have a natural transformation ψ X : X → ( X ′ ) ′ given by ψ X = Γ X , X ′ ( e X ◦ R X , X ′ ) . If C is rigid, then ψ is a natural isomorphism (see for instance [BK, Section 2.2]), and itfollows automatically that the contragredient functor permutes the simple objects of C . If C is a ribbon category, then the natural isomorphisms δ X : X → X ∗∗ defined by δ X = Γ X , X ∗ ( e X ◦ R X , X ∗ ◦ ( θ X ⊗ Id X ∗ ))have better properties (again see [BK, Section 2.2]). Definition 2.3.
Let C be a tensor category. A triple ( A , µ A , ι A ) with A an object of C and µ A : A ⊗ A → A , ι A : C → A morphisms in C is called an associative algebra if:(1) Multiplication is associative: µ A ◦ (Id A ⊗ µ A ) = µ A ◦ ( µ A ⊗ Id A ) ◦A A , A , A : A ⊗ ( A ⊗ A ) → A (2) Multiplication is unital: µ A ◦ ( ι A ⊗ Id A ) = l A : C ⊗ A → A and µ A ◦ (Id A ⊗ ι A ) = r A : A ⊗ C → A .If C is braided, we say that ( A , µ A , ι A ) is a commutative algebra if additionally:(3) Multiplication is commutative: µ A ◦ R A , A = µ A : A ⊗ A → A .We will sometimes drop the qualifiers “associative” and “commutative” when the context isclear. Remark 2.4.
In a commutative associative algebra, the right unit property µ A ◦ (Id A ⊗ ι A ) = r A is a consequence of the left unit property and the commutativity.We shall need the definition of “multiplication rules” and the corresponding multiplicationalgebra: Definition 2.5.
Let ( A , µ A , ι A ) be an algebra in a tensor category C and suppose A iscompletely reducible in C . For simple C -subobjects X , Y , Z of A we define the multiplicationrule M ZX , Y to be M ZX , Y := ( Z ⊆ Image ( µ | X ⊗ Y → A ) , . and the unital multiplication algebra of A to be the free Z -module with the set B ofinequivalent simple C -subobjects of A as basis and product X · Y := X Z ∈ B M ZX , Y Z . Direct sum completion.
We would like to work with algebras A that are actuallyinfinite direct sums of objects in C , and thus are not objects of C itself. The most naturalsetting for this is the direct sum completion of the (ribbon) category C as in [AR]. The ideais to construct an extended category C ⊕ whose objects are direct sums L s ∈ S X s of objectsin C , where S is an arbitrary index set, and whose morphisms f : L s ∈ S X s → L t ∈ T Y t aresuch that for any s ∈ S , f | X s maps to L t ∈ T ′ Y t for some finite subset T ′ ⊆ T . t was shown in [AR] that if C is a tensor category, possibly with additional structuresuch as braiding, then C ⊕ can be naturally endowed with the same structures, essentiallydefining all structure isomorphisms “componentwise.” Moreover there is a braided monoidalfunctor C → C ⊕ which is fully faithful, that is, bijective on morphisms. In effect, if we startwith a braided tensor category C with a balancing isomorphism, we can enlarge it to containarbitrary direct sums.There are three caveats:(1) First, if C is abelian, then C ⊕ is not in general abelian. However, this complicationdoes not arise if C is semisimple, since in this case C ⊕ is the category of arbitrarydirect sums of simple objects in C , which is closed under subobjects and quotients(see for instance [Ja, Section 3.5]). More generally, one could work with the smallestcategory that contains C and is closed under direct sums, kernels, and cokernels;however, we will not need this here.(2) Second, we will be working with representation categories of vertex operator alge-bras, where morphism spaces Hom( X ⊗ X , Y ) are naturally isomorphic to spaces ofintertwining operators of type (cid:0) YX X (cid:1) . We will need this same correspondence to holdin the direct sum completion.(3) Third, even if C is rigid, one can not define a coevaluation on C ⊕ . However, for ourpurposes, we will only need a contragredient functor on a subcategory of C ⊕ , as wenow explain.Assume that C is a semisimple tensor category with a contragredient functor. Let C fin ⊕ denote the full subcategory of C ⊕ whose objects contain any simple object in C with atmost finite multiplicity. Note that C fin ⊕ is not a tensor subcategory of C ⊕ unless C hasfinitely many equivalence classes of simple objects (in which case C fin ⊕ = C ). However, C fin ⊕ admits a contragredient functor in a suitable sense. If X = L s ∈ S X s is an object of C fin ⊕ ,then so is X ′ = L s ∈ S X ′ s because the contragredient functor permutes the simple objects of C . Moreover, if F : X = L s ∈ S X s → Y = L t ∈ T Y t is a morphism in C fin ⊕ , we can define F ′ : Y ′ → X ′ as follows: For any s ∈ S , t ∈ T , let F s,t : X s → Y t denote the projection onto Y t of the restriction of F to X s . Then we define F ′ by F ′ | Y ′ t = X s ∈ S F ′ s,t . To see why this sum is well defined, note that Y t , as an object of C , is the direct sum offinitely many simple objects of C with finite multiplicity. Then since X is an object of C fin ⊕ ,these finitely many simple objects can occur in only finitely many X s , and so F s,t = 0 for allbut finitely many s ∈ S . Proposition 2.6. If C is a semisimple tensor category with a contragredient functor, thenthere are natural isomorphisms Γ X , Y : Hom C ⊕ ( X ⊗ Y , ) → Hom C ⊕ ( X , Y ′ ) for X any object of C ⊕ and Y an object of C fin ⊕ .Proof. Suppose X = L s ∈ S X s and Y = L t ∈ T Y t . Then because X ⊗ Y = M ( s,t ) ∈ S × T X s ⊗ Y t , e have natural isomorphismsHom C ⊕ ( X ⊗ Y , ) ∼ = Y ( s,t ) ∈ S × T Hom C ( X s ⊗ Y t , ) ∼ = Y ( s,t ) ∈ S × T Hom C ( X s , Y ′ t ) . Now, for any tuple { F s,t } s ∈ S,t ∈ T ∈ Q ( s,t ) ∈ S × T Hom C ( X s , Y ′ t ) and s ∈ S , we must have F s,t = 0for all but finitely many t ∈ T since the finitely many simple objects occurring in X s canoccur in only finitely many Y ′ t , given that Y is an object of C fin ⊕ . This shows that in fact Y ( s,t ) ∈ S × T Hom C ( X s , Y ′ t ) ∼ = Hom C ⊕ ( X , Y ′ ) , as required. (cid:3) The details of the definitions and structures in C ⊕ are gathered in Appendix A. Since mostarguments in the following sections do not change when C is a semisimple ribbon categoryand A is an algebra in C ⊕ rather than in C , we shall frequently omit references to C ⊕ .2.3. Representation categories of an algebra object.
Now we define representationsof an algebra in a tensor category and recall some important theorems from [KO], [HKL]and [CKM]. From now on, we will assume that the tensoring functors X ⊗ • and • ⊗ X for anobject X in a tensor category C are right exact (so that in particular, these functors preservesurjections). This is needed to guarantee that the category of representations of an algebraobject, defined below, has a tensor product bifunctor. Definition 2.7.
Suppose that ( A , µ A , ι A ) is an associative algebra in C . Define Rep A tobe the category of pairs ( X , µ X ) where X ∈ Obj( C ) and µ X ∈ Hom C ( A ⊗ X , X ) satisfy thefollowing:(1) Unit property: l X = µ X ◦ ( ι A ⊗ Id X ) : C ⊗ X → X ,(2) Associativity: µ X ◦ (Id A ⊗ µ X ) = µ X ◦ ( µ A ⊗ Id X ) ◦ A A , A , X : A ⊗ ( A ⊗ X ) → X .A morphism f ∈ Hom
Rep A (( X , µ X ) , ( X , µ X )) is a morphism f ∈ Hom C ( X , X ) such that µ X ◦ (Id A ⊗ f ) = f ◦ µ X .When A is commutative, we define Rep A to be the full subcategory of Rep A containing“dyslectic” objects: those ( X , µ X ) such that µ X ◦ R X , A ◦ R A , X = µ X .Note that Rep A is the category of left A -modules. One may define the category of right A -modules analogously. It is easy to show that if ( X , µ X ) is in Rep A , then ( X , µ X ◦ R X , A )and ( X , µ X ◦ R − A , X ) are right modules for the opposite algebras ( A , µ A ◦ R A , A , ι A ) and ( A , µ A ◦R − A , A , ι A ), respectively. Note that when A is commutative, both opposite algebras are equalto A and the dyslectic modules X (objects of Rep A ) are precisely those for which the tworight A -module structures on V coincide.Clearly, ( A , µ A ) is both a left and a right A -module, and an object of Rep A when A iscommutative.We have an induction functor F : C →
Rep A given on objects by F ( W ) = A ⊗ W for objects W in C and on morphisms by F ( f ) = Id A ⊗ f for f : W → W in C . Note that if A is an algebra in C ⊕ , we will still take C as the domainof our induction functor to Rep A . Critically, the induction functor F satisfies Frobenius eciprocity [KO, CKM], that is, it is left adjoint to the forgetful functor G from Rep A to C : there is a natural isomorphismHom Rep A ( F ( W ) , X ) ∼ = Hom C ( W , G ( X )) (2.2)for objects W in C and X in Rep A . Under this isomorphism, f ∈ Hom
Rep A ( F ( W ) , X ) mapsto the composition W → ⊗ W ι A ⊗ Id W −−−−→ A ⊗ W f −→ X , and g ∈ Hom C ( W , G ( X )) maps to the composition A ⊗ W Id A ⊗ g −−−→ A ⊗ X µ X −→ X . When A is commutative, the category Rep A is a tensor category with tensor product ⊗ A and unit object A , and the subcategory Rep A is a braided tensor category (see for example[KO, CKM]). Since ⊗ A is defined as the cokernel of a certain morphism in C , if A is analgebra in C ⊕ , we assume that C is semisimple to guarantee C ⊕ is abelian. Crucially, theinduction functor is monoidal [KO, CKM], that is, there are natural Rep A -isomorphisms f W , W : F ( W ⊗ W ) ∼ −→ F ( W ) ⊗ A F ( W ) (2.3)which together with the Rep A -isomorphism r A : F ( ) ∼ −→ A are suitably compatible withthe unit and associativity isomorphisms in C and Rep A .The following lemma in the case that C is rigid amounts to part of [KO, Theorem 1.15],but here we assume only that C has a contragredient functor because we will need to applythe result in C ⊕ . Lemma 2.8. If C has a contragredient functor and ( X , µ X ) is an object of Rep A , then ( X ′ , µ X ′ ) is a right A -module, where µ X ′ : X ′ ⊗ A → X ′ is given by µ X ′ = Γ X ′ ⊗ A , X (cid:0) e X ◦ (Id X ′ ⊗ µ X ) ◦ A − X ′ , A , X (cid:1) . Proof.
To prove the unit property of µ X ′ , we use the commutative diagramHom C (( X ′ ⊗ A ) ⊗ X , ) F F ◦ ((Id ′ X ⊗ ι A ) r − X ′ ⊗ Id X ) (cid:15) (cid:15) Hom C ( X ′ ⊗ A , X ′ ) Γ − X ′⊗ A , X o o G G ◦ (Id X ′ ⊗ ι A ) ◦ r − X ′ (cid:15) (cid:15) Hom C ( X ′ ⊗ X , ) Hom C ( X ′ , X ′ ) Γ − X ′ , X o o given by the naturality of Γ. Applying both compositions to µ X ′ , and then using the definitionof µ X ′ , the naturality of the associativity isomorphisms, the triangle axiom, and the unitproperty for X , we getΓ − X ′ , X ( µ X ′ ◦ (Id X ′ ⊗ ι A ) ◦ r − X ′ ) = Γ − X ′ ⊗ A , X ( µ X ′ ) ◦ ((Id X ′ ⊗ ι A ) ⊗ Id X ) ◦ ( r − X ′ ⊗ Id X )= e X ◦ (Id X ′ ⊗ µ X ) ◦ A − X ′ , A , X ◦ ((Id X ′ ⊗ ι A ) ⊗ Id X ) ◦ ( r − X ′ ⊗ Id X )= e X ◦ (Id X ′ ⊗ µ X ) ◦ (Id X ′ ⊗ ( ι A ⊗ Id X )) ◦ A − X ′ , , X ◦ ( r − X ′ ⊗ Id X )= e X ◦ (Id X ′ ⊗ µ X ) ◦ (Id X ′ ⊗ ( ι A ⊗ Id X )) ◦ (Id X ′ ⊗ l − X )= e X . We conclude that µ X ′ ◦ (Id X ′ ⊗ ι A ) ◦ r − X ′ = Γ X ′ , X ( e X ) = Id X ′ s required.To prove the associativity of µ X ′ , we first use the commutative diagramHom C ( X ′ ⊗ A , X ′ ) Γ − X ′⊗ A , X / / G G ◦ ( µ X ′ ⊗ Id A ) (cid:15) (cid:15) Hom C (( X ′ ⊗ A ) ⊗ X , ) F F ◦ (( µ X ′ ⊗ Id A ) ⊗ Id X ) (cid:15) (cid:15) Hom C (( X ′ ⊗ A ) ⊗ A , X ′ ) Γ − X ′⊗ A ) ⊗ A , X / / Hom C ((( X ′ ⊗ A ) ⊗ A ) ⊗ X , )given by the naturality of Γ. Applying both compositions to µ X ′ and then using the definitionof µ X ′ and the naturality of the associativity isomorphisms, we getΓ − X ′ ⊗ A ) ⊗ A , X ( µ X ′ ◦ ( µ X ′ ⊗ Id A )) = Γ − X ′ ⊗ A , X ( µ X ′ ) ◦ (( µ X ′ ⊗ Id A ) ⊗ Id X )= e X ◦ (Id X ′ ⊗ µ X ) ◦ A − X ′ , A , X ◦ (( µ X ′ ⊗ Id A ) ⊗ Id X )= e X ◦ ( µ X ′ ⊗ Id X ) ◦ (Id X ′ ⊗ A ⊗ µ X ) ◦ A − X ′ ⊗ A , A , X . (2.4)Now, the naturality of Γ implies e X ◦ ( µ X ′ ⊗ Id X ) = Γ − X ′ , X (Id X ′ ) ◦ ( µ X ′ ⊗ Id X )= Γ − X ′ ⊗ A , X (Id X ′ ◦ µ X ′ ) = e X ◦ (Id X ′ ⊗ µ X ) ◦ A − X ′ , A , X . Putting this back into (2.4) and using the naturality of the associativity isomorphisms, theassociativity of µ X the pentagon axiom, the definition of µ X ′ , and the naturality of Γ, we getΓ − X ′ ⊗ A ) ⊗ A , X ( µ X ′ ◦ ( µ X ′ ⊗ Id A ))= e X ◦ (Id X ′ ⊗ µ X ) ◦ A − X ′ , A , X ◦ (Id X ′ ⊗ A ⊗ µ X ) ◦ A − X ′ ⊗ A , A , X = e X ◦ (Id X ′ ⊗ µ X ) ◦ (Id X ′ ⊗ (Id A ⊗ µ X )) ◦ A − X ′ , A , A ⊗ X ◦ A − X ′ ⊗ A , A , X = e X ◦ (Id X ′ ⊗ µ X ) ◦ (Id X ′ ⊗ ( µ A ⊗ Id X )) ◦ (Id X ′ ⊗ A A , A , X ) ◦ A − X ′ , A , A ⊗ X ◦ A − X ′ ⊗ A , A , X = e X ◦ (Id X ′ ⊗ µ X ) ◦ (Id X ′ ⊗ ( µ A ⊗ Id X )) ◦ A − X ′ , A ⊗ A , X ◦ ( A − X ′ , A , A ⊗ Id X )= e X ◦ (Id X ′ ⊗ µ X ) ◦ A − X ′ , A , X ◦ ((Id X ′ ⊗ µ A ) ⊗ Id X ) ◦ ( A − X ′ , A , A ⊗ Id X )= Γ − X ′ ⊗ A , X ( µ X ′ ) ◦ ((Id X ′ ⊗ µ A ) ⊗ Id X ) ◦ ( A − X ′ , A , A ⊗ Id X )= Γ − X ′ ⊗ A ) ⊗ A , X ( µ X ′ ◦ (Id X ′ ⊗ µ A ) ◦ A − X ′ , A , A ) . Applying Γ ( X ′ ⊗ A ) ⊗ A , X to both sides, we conclude µ X ′ ◦ ( µ X ′ ⊗ Id A ) = µ X ′ ◦ (Id X ′ ⊗ µ A ) ◦ A − V ′ , A , A , which is the associativity of µ X ′ . (cid:3) Although in this paper we will mostly be concerned with commutative algebras, we willprove some results for superalgebras, which are associative algebras that in particular satisfy µ A ◦ R A , A = µ A . For such an associative algebra, we have a single opposite algebra A op =( A , µ A ◦ R A , A , ι A ). Thus in light of the preceding lemma and Proposition 2.6, we immediatelyhave: Corollary 2.9.
Assume that C is a braided tensor category with contragredient functor andthat either: (1) A is an associative algebra in C , or C is semisimple and A is an associative algebra in C fin ⊕ .If µ A ◦ R A , A = µ A , then ( A ′ , µ + A ′ = µ A ′ ◦ R A , A ′ ) and ( A ′ , µ − A ′ = µ A ′ ◦ R − A ′ , A ) are objects of Rep A op . Now the following lemma in the case that A is an algebra in C is essentially a special caseof [KO, Theorem 1.17.3], but again we will need the result for algebras in C fin ⊕ : Lemma 2.10.
Assume that C is a rigid braided tensor category and that either: (1) A is an associative algebra in C , or (2) C is semisimple and A is an associative algebra in C fin ⊕ .If µ A ◦ R A , A = µ A , then the morphism ψ A of Remark 2.2 is an isomorphism in Rep A from ( A , µ A ) to ( A ∗∗ , µ A ∗∗ ◦ R A , A ∗∗ ) , where the right A op -module structure µ A ∗∗ is obtained usingthe left A op -module structure µ + A ∗ on A ∗ .Proof. First observe that ( A ∗∗ , µ A ∗∗ ◦ R A , A ∗∗ ) is actually an object of Rep A because ourassumption µ A ◦ R A , A = µ A implies that ( A op ) op = A .If A is an algebra in C , then ψ A is an isomorphism in C because C is rigid. If on the otherhand A = L s ∈ S A s is an algebra in C fin ⊕ , then using the definition ψ A = Γ A , A ∗ ( e A ◦ R A , A ∗ ) , where Γ A , A ∗ is the natural isomorphism of Proposition 2.6, together with the natural isomor-phism Hom C ⊕ ( A , A ∗∗ ) ∼ = Y ( s,t ) ∈ S × S Hom C ( A s , A ∗∗ t ) , we can identify ψ A with the tuple { δ s,t ψ A s } ( s,t ) ∈ S × S ∈ Y ( s,t ) ∈ S × S Hom C ( A s , A ∗∗ t ) . Now since C is rigid, each ψ A s is an isomorphism in C , and so ψ A is an isomorphism in C ⊕ .It remains to show that ψ A is actually a morphism in Rep A . We need to show that µ A ∗∗ ◦ R A , A ∗∗ ◦ (Id A ⊗ ψ A ) = ψ A ◦ µ A . By the naturality of the braiding isomorphisms and the assumption µ A ◦ R A , A = µ A , this isequivalent to showing µ A ∗∗ ◦ ( ψ A ⊗ Id A ) = ψ A ◦ µ A ◦ R A , A (that is, ψ A is a homomorphism of right A op -modules). For this, we first note a general factabout contragredient functors: if f : W ⊗ X → is a morphism in C , then Γ W , X ( f ) is theunique morphism in Hom C ( W , X ′ ) such that e X ◦ (Γ W , X ( f ) ⊗ Id X ) = f . This follows fromapplying both compositions in the commutative diagramHom C ( X ′ ⊗ X , ) Γ X ′ , X / / F F ◦ (Γ( f ) ⊗ Id X ) (cid:15) (cid:15) Hom C ( X ′ , X ′ ) G G ◦ Γ( f ) (cid:15) (cid:15) Hom C ( W ⊗ X , ) Γ W , X / / Hom C ( W , X ′ ) o e X = Γ − X ′ , X (Id X ′ ). In the cases Γ W , X ( f ) = µ A , A ∗∗ and Γ W , X ( f ) = ψ A , we see that µ A , A ∗∗ : A ∗∗ ⊗ A → A ∗∗ is the unique morphism such that e A ∗ ◦ ( µ A ∗∗ ⊗ Id A ∗ ) = e A ∗ ◦ (Id A ∗∗ ⊗ µ + A ∗ ) ◦ A − A ∗∗ , A , A ∗ and ψ A : A → A ∗∗ is the unique morphism such that e A ∗ ◦ ( ψ A ⊗ Id A ∗ ) = e A ◦ R A , A ∗ . Using these relations together with the naturality of the associativity isomorphisms, weobtain the commutative diagram( A ⊗ A ) ⊗ A ∗ A − A , A , A ∗ / / ( ψ A ⊗ Id A ) ⊗ Id A ∗ (cid:15) (cid:15) A ⊗ ( A ⊗ A ∗ ) Id A ⊗R A , A ∗ / / ψ A ⊗ Id A ⊗ A ∗ (cid:15) (cid:15) A ⊗ ( A ∗ ⊗ A ) ψ A ⊗ Id A ∗⊗ A (cid:15) (cid:15) Id A ⊗ µ A ∗ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ( A ∗∗ ⊗ A ) ⊗ A ∗ A − A ∗∗⊗ A ⊗ A ∗ / / µ A ∗∗ ⊗ Id A ∗ * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ A ∗∗ ⊗ ( A ⊗ A ∗ ) Id A ∗∗ ⊗R A , A ∗ / / Id A ∗∗ ⊗ µ + A ∗ * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ A ∗∗ ⊗ ( A ∗ ⊗ A ) Id A ∗∗ ⊗ µ A ∗ (cid:15) (cid:15) A ⊗ A ∗ ψ A ⊗ Id A ∗ u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ R A , A ∗ (cid:15) (cid:15) A ∗∗ ⊗ A ∗ e A ∗ * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ A ∗∗ ⊗ A ∗ e A ∗ (cid:15) (cid:15) A ∗ ⊗ A e A u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Now we can use the naturality of the braiding isomorphisms and the hexagon axioms torewrite the outer composition on the top and right sides of the diagram as( A ⊗ A ) ⊗ A ∗ A − A , A , A ∗ −−−−→ A ⊗ ( A ⊗ A ∗ ) Id A ⊗R A , A ∗ −−−−−−→ A ⊗ ( A ∗ ⊗ A ) A A , A ∗ , A −−−−→ ( A ⊗ A ∗ ) ⊗ A R A , A ∗ ⊗ Id A −−−−−−→ ( A ∗ ⊗ A ) ⊗ A A − A ∗ , A , A −−−−→ A ∗ ⊗ ( A ⊗ A ) Id A ∗ ⊗R A , A −−−−−−→ A ∗ ⊗ ( A ⊗ A ) A A ∗ , A , A −−−−→ ( A ∗ ⊗ A ) ⊗ A µ A ∗ ⊗ Id A −−−−−→ A ∗ ⊗ A e A −→ . Next we use the hexagon axioms to rewrite the first five arrows and use the relation e A ◦ ( µ A ∗ ⊗ Id A ) = e A ◦ (Id A ∗ ⊗ µ A ) ◦ A − A ∗ , A , A to obtain( A ⊗ A ) ⊗ A ∗ R A ⊗ A , A ∗ −−−−−→ A ∗ ⊗ ( A ⊗ A ) Id A ∗ ⊗R A , A −−−−−−→ A ∗ ⊗ ( A ⊗ A ) Id A ∗ ⊗ µ A −−−−−→ A ∗ ⊗ A e A −→ . The naturality of the braiding now imply this is( A ⊗ A ) ⊗ A ∗ R A , A −−→ ( A ⊗ A ) ⊗ A ∗ µ A ⊗ Id A ∗ −−−−−→ A ⊗ A ∗ R A , A ∗ −−−→ A ∗ ⊗ A e A −→ , which is Γ − A , A ∗ ( ψ A ) ◦ (( µ A ◦ R A , A ) ⊗ Id A ∗ ).Our calculations have now shown that e A ∗ ◦ ( µ A ∗∗ ◦ ( ψ A ⊗ Id A ) ⊗ Id A ∗ ) = Γ − A , A ∗ ( ψ A ) ◦ (( µ A ◦ R A , A ) ⊗ Id A ∗ )= Γ − A ⊗ A , A ∗ ( ψ A ◦ µ A ◦ R A , A ) , where we have used the naturality of Γ for the second equality. Applying Γ A ⊗ A , A ∗ to bothsides then yields the desired equality µ A ∗∗ ◦ ( ψ A ⊗ Id A ) = ψ A ◦ µ A ◦ R A , A . (cid:3) The main reason we need the preceding lemma is the following corollary: orollary 2.11. Assume that C is a rigid braided tensor category and that either: (1) A is an associative algebra in C , or (2) C is semisimple and A is an associative algebra in C fin ⊕ .If µ A ◦ R A , A = µ A and A is simple as an object of Rep A , then ( A ∗ , µ A ∗ ) is a simple right A -module.Proof. We need to show that any right A -module inclusion (equivalently, Rep A op -inclusion) i : V → A ∗ is either 0 or an isomorphism. In fact, the cokernel c : A ∗ → coker i is alsomorphism in Rep A op (see for instance [KO, Lemma 1.4] or [CKM, Theorem 2.9]). It isstraightforward to show that the dual c ∗ : (coker i ) ∗ → A ∗∗ is then a right A op -module homomorphism (equivalently, a morphism in Rep A ), and it isinjective. But A ∗∗ is simple in Rep A since it is isomorphic to A . Therefore c ∗ is either 0 oran isomorphism, and the same then holds for c and i . (cid:3) The center of a tensor category.
The center Z ( C ) of a tensor category C is animportant construction we will use for studying the commutativity of algebras in C . Definition 2.12.
Let C be a tensor category. The center Z ( C ) is the category whose objectsare pairs ( X , γ X ) where X ∈ Obj( C ) and γ X = { γ XM : M ⊗ X → X ⊗ M | M ∈ C} is a familyof isomorphisms in C , called a half-braiding , that are natural in the sense that M ⊗ X γ XM / / f ⊗ Id X (cid:15) (cid:15) X ⊗ M Id X ⊗ f (cid:15) (cid:15) N ⊗ X γ XN / / X ⊗ N . commutes for all f in Hom C ( M , N ) and such that( Y ⊗ Z ) ⊗ X γ XY ⊗ Z / / X ⊗ ( Y ⊗ Z ) A X , Y , Z * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ Y ⊗ ( Z ⊗ X ) A Y , Z , X ♥♥♥♥♥♥♥♥♥♥♥♥ Id Y ⊗ γ XZ ( ( PPPPPPPPPPPP ( X ⊗ Y ) ⊗ Z (Hexagon 1) Y ⊗ ( X ⊗ Z ) A Y , X , Z / / ( Y ⊗ X ) ⊗ Z γ XY ⊗ Id Z ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ commutes for all objects Y , Z . A morphism in Z ( C ) from ( X , γ X ) to ( Y , γ Y ) is a C -morphism f from X to Y satisfying commutativity of the following diagram for all M ∈ Obj( C ). M ⊗ X Id M ⊗ f / / γ XM (cid:15) (cid:15) M ⊗ Y γ YM (cid:15) (cid:15) X ⊗ M f ⊗ Id M / / Y ⊗ M . (2.5) e have a forgetful functor I : Z ( C ) → C with I ( X , γ X ) = X for an object ( X , γ X ) and I ( f ) = f for a morphism f .A basic property of half-braidings that we will use is the following: Lemma 2.13. If γ X is a half-braiding, then γ X = r − X ◦ l X .Proof. Using the naturality of γ X , Hexagon 1, and properties of the unit in C , the followingdiagram commutes: ⊗ X γ X / / X ⊗ ⊗ ( ⊗ X ) l ⊗ X ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ A , , X / / Id ⊗ γ X * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ ( ⊗ ) ⊗ X ( l = r ) ⊗ Id X O O γ X ⊗ / / X ⊗ ( ⊗ ) Id X ⊗ ( l = r ) O O A X , , / / ( X ⊗ ) ⊗ r X ⊗ j j ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ ⊗ ( X ⊗ ) A , X , / / ( ⊗ X ) ⊗ γ X ⊗ Id ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ That is, γ X = r X ⊗ ◦ ( γ X ⊗ Id ) ◦ A , X , ◦ (Id ⊗ γ X ) ◦ l − ⊗ X = γ X ◦ r ⊗ X ◦ A , X , ◦ l − X ⊗ ◦ γ X , using also the naturality of the unit isomorphisms. So γ X = l X ⊗ ◦ A − , X , ◦ r − ⊗ X = l X ⊗ ◦ (Id ⊗ r − X )= r − X ◦ l X , by properties of the unit isomorphisms. (cid:3) The center is a tensor category with tensor product( X , γ X ) ⊗ ( Y , γ Y ) = (cid:0) X ⊗ Y , γ X ⊗ Y (cid:1) , γ X ⊗ Y := A X , Y , • (cid:0) Id X ⊗ γ Y (cid:1) A − X , • , Y (cid:0) γ X ⊗ Id Y (cid:1) A • , X , Y . (2.6)The definition of γ X ⊗ Y is precisely saying that the diagram M ⊗ ( X ⊗ Y ) γ X ⊗ YM / / ( X ⊗ Y ) ⊗ M A − X , Y , M * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ ( M ⊗ X ) ⊗ Y A − M , X , Y ♥♥♥♥♥♥♥♥♥♥♥♥ γ XM ⊗ Id Y ( ( PPPPPPPPPPPP X ⊗ ( Y ⊗ M ) (Hexagon 2)( X ⊗ M ) ⊗ Y A − X , M , Y / / X ⊗ ( M ⊗ Y ) Id X ⊗ γ YM ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ commutes for all objects M in C . The unit of the center is Z ( C ) = ( C , r − ◦ l ) with r and l the right and left unit constraints in C . he two commutative diagrams (Hexagon 1 and 2) together with the naturality of half-braidings ensure that for objects ( X , γ X ) and ( Y , γ Y ) in Z ( C ), the C -isomorphism R ( X ,γ X ) , ( Y ,γ Y ) := γ YX : ( X ⊗ Y , γ X ⊗ Y ) → ( Y ⊗ X , γ Y ⊗ X )is actually a morphism in Z ( C ) and defines a braiding on the center. If C is already a braidedtensor category, then we have two tensor functors from C to Z ( C ): F : X ( X , R • , X ) , f f and F rev : X ( X , R − X , • ) , f f. Note that composing either F or F rev with the forgetful functor I : Z ( C ) → C yields theidentity functor on C . In fact, F is an example of a central functor structure on Id C . Ingeneral, a tensor functor F : C → M where C is a braided tensor category is a centralfunctor if there is a braided tensor functor G : C → Z ( M ) such that the diagram C F / / G " " ❊❊❊❊❊❊❊❊❊ MZ ( M ) I ; ; ✇✇✇✇✇✇✇✇✇ commutes. In the above example, F rev is a central functor structure on Id C rev , where C rev equals C as a tensor category but has reversed braidings: R rev X , Y = R − Y , X .We can slightly generalize the above central functor structures on the identity of a braidedtensor category in the following way. Suppose we have a fully faithful tensor functor F : C →M , with C braided, so that F is an equivalence of tensor categories between C and F ( C ) (thefull subcategory of M consisting of objects isomorphic to some F ( X ) for X an object of C ).Then we can choose a functor F ′ : F ( C ) → C and natural isomorphisms η : Id F ( C ) → F ◦ F ′ , h : F ′ ◦ F → Id C which satisfy F ( h X ) = η − F ( X ) for any object X in C (see for instance the proof of [Ka, Proposition XI.1.5]). Then F : C →F ( C ) is a central functor with extension G : C → Z ( F ( C )) defined as follows: G : X ( F ( X ) , γ F ( X ) ) , f
7→ F ( f )where for an object Y of F ( C ), γ F ( X ) Y is the composition Y ⊗ F ( X ) η Y ⊗ Id F ( X ) −−−−−−→F ( F ′ ( Y )) ⊗ F ( X ) ∼ = −→ F ( F ′ ( Y ) ⊗ X ) F ( R F′ ( Y ) , X ) −−−−−−−→ F ( X ⊗ F ′ ( Y )) ∼ = −→ F ( X ) ⊗ F ( F ′ ( Y )) Id X ⊗ η − Y −−−−−→ F ( X ) ⊗ Y . The tensor functor F : C rev → F ( C ) is also central with a braided extension given by G rev : X ( F ( X ) , ( γ rev ) F ( X ) ) , f
7→ F ( f ) , where ( γ rev ) F ( X ) Y is defined similarly to γ F ( X ) Y , except that we use R − X , F ′ ( Y ) instead of R F ′ ( Y ) , X . Definition 2.14.
Given a braided tensor category C , and a full braided tensor subcategory B , the centralizer B ′ is the full subcategory of C such that X ∈ Obj( C ) is an object of B ′ ifand only if R Y , X ◦ R X , Y = Id X ⊗ Y for all Y ∈ Obj( B ). t is easy to see from the hexagon axioms that B ′ is a braided tensor subcategory of C . Thefollowing proposition is essemtially [M¨u, Proposition 7.3]; although it is only stated therefor fusion categories, it is clear from the proof that no finiteness or semisimplicity conditionsare necessary. Proposition 2.15.
Let C be a braided tensor category and suppose F : C → M is a fullyfaithful tensor functor. Then with the braided tensor functors G : C → Z ( F ( C )) and G rev : C rev → Z ( F ( C )) defined as above, we have G ( C ) ′ = G rev ( C rev ) and G rev ( C rev ) ′ = G ( C ) . Proof.
An object ( F ( X ) , g F ( X ) ) of Z ( F ( C )) is an object of G ( C ) ′ precisely when g F ( X ) F ( Y ) ◦ γ F ( Y ) F ( X ) = Id F ( X ) ⊗F ( Y ) for all objects Y in C . Since g F ( X ) is natural, this occurs precisely when, for all objects Z in F ( C ), g F ( X ) Z = (Id F ( X ) ⊗ η − Z ) ◦ g F ( X ) F ( F ′ ( Z )) ◦ ( η Z ⊗ Id F ( X ) ) = (Id F ( X ) ⊗ η − Z ) ◦ ( γ F ( F ′ ( Z )) F ( X ) ) − ◦ ( η Z ⊗ Id F ( X ) ) . By definition, this means g F ( X ) Z is the composition Z ⊗ F ( X ) η Z ⊗ η F ( X ) −−−−−→ F ( F ′ ( Z )) ⊗ F ( F ′ ( F ( X ))) ∼ = −→ F ( F ′ ( Z ) ⊗ F ′ ( F ( X ))) F ( R − F′ ( F ( X )) , F′ ( Z ) ) −−−−−−−−−−→ F ( F ′ ( F ( X )) ⊗ F ′ ( Z )) ∼ = −→ F ( F ′ ( F ( X ))) ⊗ F ( F ′ ( Z )) η − F ( X ) ⊗ η − Z −−−−−−→ F ( X ) ⊗ Z . However, because η − F ( X ) = F ( h X ) and because the central isomorphisms in the compositionare natural, we actually have g F ( X ) Z equal to the composition Z ⊗ F ( X ) η Z ⊗ Id F ( X ) −−−−−−→F ( F ′ ( Z )) ⊗ F ( X ) ∼ = −→ F ( F ′ ( Z ) ⊗ X ) F ( R − X , F′ ( Z ) ) −−−−−−−→ F ( X ⊗ F ′ ( Z )) ∼ = −→ F ( X ) ⊗ F ( F ′ ( Z )) Id X ⊗ η − Z −−−−−→ F ( X ) ⊗ Z , which is precisely ( γ rev ) F ( X ) Z . This shows that G ( C ) ′ = G rev ( C rev ), and the proof that G rev ( C rev ) ′ = G ( C ) is the same. (cid:3) From braid-reversed equivalences to algebras
In this section we construct the canonical algebra associated to a tensor category C , payingclose attention to the assumptions on C needed to ensure the existence and commutativityof this algebra. In particular, we will see that when C is rigid, braided, and semisimple buthas infinitely many equivalence classes of simple objects, then the canonical algebra may beconstructed in C ⊕ .3.1. Associative algebras.
For this subsection, we take C to be an F -linear (abelian) tensorcategory, not necessarily braided or finite, where F is a field. (For our purposes in this paper,we may take F = C .) We will construct associative algebras in C associated to a C -modulecategory M . Although this construction seems to be well known (see for example [EGNO,Exercise 7.9.9]), we could not find a complete proof in the literature.Recall that in a C -module category M we have natural associativity isomorphisms A X , X , M : X ⊗ ( X ⊗ M ) → ( X ⊗ X ) ⊗ M . or X , X ∈ Obj( C ) , M ∈ Obj( M ). For every pair M , M ∈ Obj( M ), we have a contravariantfunctor G M , M : C → V ec F X Hom M ( X ⊗ M , M ) for X ∈ Obj( C ) f ( g g ◦ ( f ⊗ Id M )) for f ∈ Hom C ( X , Y ) , g ∈ Hom M ( Y ⊗ M , M ) . Assume that G M , M is representable, which means that there exists Hom( M , M ), perhapsin a suitable completion of C , called the internal hom of M and M such that there arenatural isomorphisms λ X ( M , M ) : Hom M ( X ⊗ M , M ) ∼ = −−−→ Hom C ( X , Hom( M , M )) . (3.1)So, for f : X → Y with X , Y ∈ Obj( C ), we have the following commuting diagram:Hom M ( X ⊗ M , M ) λ X / / Hom C ( X , Hom( M , M ))Hom M ( Y ⊗ M , M ) λ Y / / g g ◦ ( f ⊗ Id M ) O O Hom C ( Y , Hom( M , M )) . h h ◦ f O O Remark 3.1.
By [EGNO, Corollary 1.8.11, Section 7.9], G M , M is representable if C is afinite (multi)tensor category. If C is not finite, then [EGNO, Section 7.9] states that internalhoms still exist as ind-objects of the completion ind − C . In the setting of vertex operatoralgebras, certain internal homs were constructed in [Li2] as weak modules (and thus notnecessarily objects of C itself).Fix an object M ∈ Obj( M ) and abbreviate λ X := λ X ( M , M ) and A := Hom( M , M ). For i = 1 , ,
3, we will be using the notation ϕ i for a morphism in Hom M ( X i ⊗ M , M ), where X i ∈ Obj( C ). For X , X ∈ Obj( C ), we have a linear map ν X , X : Hom M ( X ⊗ M , M ) ⊗ F Hom M ( X ⊗ M , M ) → Hom M (( X ⊗ X ) ⊗ M , M )under which ϕ ⊗ F ϕ is sent to the composition( X ⊗ X ) ⊗ M A − X , X , M −−−−−→ X ⊗ ( X ⊗ M ) Id X ⊗ ϕ −−−−−→ X ⊗ M ϕ −−→ M . Then the natural family of isomorphisms { λ X } induces a natural family of linear maps µ X , X : Hom C ( X , A ) ⊗ F Hom C ( X , A ) → Hom C (( X ⊗ X ) , A ) ,λ X ( ϕ ) ⊗ F λ X ( ϕ ) λ X ⊗ X ( ν X , X ( ϕ ⊗ F ϕ )) . Note that ν is a natural transformation of contravariant bifunctors M × M → V ec F and µ isa natural transformation of contravariant bifunctors C × C → V ec F . By definition, we have: λ X ⊗ X ◦ ν X , X = µ X , X ◦ ( λ X ⊗ F λ X ) . (3.2) onsider the diagram(( X ⊗ X ) ⊗ X ) ⊗ M A − X , X , X ⊗ Id M / / A − X ⊗ X , X , M (cid:15) (cid:15) ( X ⊗ ( X ⊗ X )) ⊗ M A − X , X ⊗ X , M (cid:15) (cid:15) ( X ⊗ X ) ⊗ ( X ⊗ M ) Id X ⊗ X ⊗ ϕ (cid:15) (cid:15) A − X , X , X ⊗ M , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ X ⊗ (( X ⊗ X ) ⊗ M ) Id X ⊗A − X , X , M (cid:15) (cid:15) ( X ⊗ X ) ⊗ M A − X , X , M ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ X ⊗ ( X ⊗ ( X ⊗ M )) Id X ⊗ (Id X ⊗ ϕ ) u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ X ⊗ ( X ⊗ M ) ϕ ◦ (Id X ⊗ ϕ ) (cid:15) (cid:15) M which commutes due to naturality of associativity and the pentagon axiom for M . Bynaturality of both λ and associativity we then have the corresponding commutative diagram( X ⊗ X ) ⊗ X A − X , X , X / / LHS( ϕ ,ϕ ,ϕ ) & & ▼▼▼▼▼▼▼▼▼▼▼▼ X ⊗ ( X ⊗ X ) RHS( ϕ ,ϕ ,ϕ ) x x qqqqqqqqqqqq A (3.3)where (leaving out identity morphisms for readability)LHS( ϕ , ϕ , ϕ ) = λ ( X ⊗ X ) ⊗ X (cid:0) ϕ ◦ ϕ ◦ ϕ ◦ A − X , X , X ⊗ M ◦ A − X ⊗ X , X , M (cid:1) = λ ( X ⊗ X ) ⊗ X (cid:0) ϕ ◦ ϕ ◦ A − X , X , M ◦ ϕ ◦ A − X ⊗ X , X , M (cid:1) , RHS( ϕ , ϕ , ϕ ) = λ X ⊗ ( X ⊗ X ) (cid:0) ϕ ◦ ϕ ◦ ϕ ◦ A − X , X ⊗ M ◦ A − X , X ⊗ X , M (cid:1) . We now define m := µ A , A (Id A ⊗ C Id A ) = λ A ⊗ A (cid:0) λ − A (Id A ) ◦ λ − A (Id A ) ◦ A − A , A , M (cid:1) (3.4)Given any f : X → A and f : X → A , we get the following commuting diagram:Hom( X ⊗ X , A ) Hom( A ⊗ A , A ) o o Hom( X , A ) ⊗ F Hom( X , A ) µ X , X O O Hom( A , A ) ⊗ F Hom( A , A ) µ A , A O O o o (3.5)Tracing the image of Id A ⊗ F Id A ∈ Hom( A , A ) ⊗ F Hom(
A, A ) gets us the following commutativediagram: X ⊗ X µ X , X ( f ⊗ C f ) / / f ⊗ f % % ❑❑❑❑❑❑❑❑❑❑ AA ⊗ A m < < ②②②②②②②②② . (3.6) aking X = A ⊗ A , X = A , f = m , f = Id A , this diagram implies that m ◦ ( m ⊗ Id A ) = µ A ⊗ A , A ( m ⊗ F Id A )= µ A ⊗ A , A (cid:0) λ A ⊗ A (cid:0) λ − A (Id A ) ◦ λ − A (Id A ) ◦ A − A , A , M (cid:1) ⊗ F Id A (cid:1) = λ ( A ⊗ A ) ⊗ A (cid:0) λ − A ⊗ A (cid:0) λ A ⊗ A (cid:0) λ − A (Id A ) ◦ λ − A (Id A ) ◦ A − A , A , M (cid:1)(cid:1) ◦ λ − A (Id A ) ◦ A − A ⊗ A , A , M (cid:1) = λ ( A ⊗ A ) ⊗ A (cid:0) λ − A (Id A ) ◦ λ − A (Id A ) ◦ A − A , A , M ◦ λ − A (Id A ) ◦ A − A ⊗ A , A , M (cid:1) = LHS( λ − A (Id A ) , λ − A (Id A ) , λ − A (Id A )) . Analogously m ◦ (Id A ⊗ m ) = µ A , A ⊗ A (Id A ⊗ F m )= µ A , A ⊗ A (cid:0) Id A ⊗ F λ A ⊗ A (cid:0) λ − A (Id A ) ◦ λ − A (Id A ) ◦ A − A , A , M (cid:1)(cid:1) = λ A ⊗ ( A ⊗ A ) (cid:0) λ − A (Id A ) ◦ λ − A (Id A ) ◦ λ − A (Id A ) ◦ A − A , A , M ◦ A − A , A ⊗ A , M (cid:1) = RHS( λ − A (Id A ) , λ − A (Id A ) , λ − A (Id A )) , so that this computation together with (3.3) implies that all triangles of the diagram com-mute: ( A ⊗ A ) ⊗ A A − A , A , A / / m ⊗ Id A (cid:15) (cid:15) LHS (cid:28) (cid:28) ✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾ A ⊗ ( A ⊗ A ) Id A ⊗ m (cid:15) (cid:15) RHS (cid:2) (cid:2) ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ A ⊗ A m % % ▲▲▲▲▲▲▲▲▲▲▲ A ⊗ A m y y rrrrrrrrrrr A (3.7)with LHS = LHS( λ − A (Id A ) , λ − A (Id A ) , λ − A (Id A )), RHS = RHS( λ − A (Id A ) , λ − A (Id A ) , λ − A (Id A )).Thus the multiplication m : A ⊗ A → A is associative.The natural candidate for a unit morphism is ι A : → A is λ ( l M ) , where l M : ⊗ M → M is the left unit isomorphism for M . We have: m ◦ ( ι A ⊗ Id A ) = µ , A ( ι A ⊗ F Id A ) = λ ⊗ A (cid:0) λ − ( λ ( l M )) ◦ λ − A (Id A ) ◦ A − , A , M (cid:1) = λ ⊗ A (cid:0) l M ◦ λ − A (Id A ) ◦ A − , A , M (cid:1) = λ ⊗ A (cid:0) λ − A (Id A ) ◦ ( l A ⊗ Id M ) (cid:1) = Id A ◦ l A = l A , (3.8)where the first equality follows by (3.6), fourth by properties of unit isomorphisms and fifthby naturality of λ . We conclude Theorem 3.2.
Let C be a multitensor category, M a left C -module category, and M an objectof M such that the functor G M , M is representable. Then with the natural isomorphisms λ defined by (3.1) , A := Hom( M , M ) together with left unit ι A = λ ( l M ) and multiplication m = λ A ⊗ A (cid:0) λ − A (Id A ) ◦ λ − A (Id A ) ◦ A − A , A , M (cid:1) is a left-unital associative algebra in C . Commutative algebras.
Now taking C to be a braided tensor category, we will findconditions under which the algebra A of the previous subsection is commutative. For this,we will take M of the previous subsection to be itself a tensor category with tensor unit M ,and we will consider the algebra A = Hom( M , M ).We assume there are natural associativity isomorphisms A X , M , M : X ⊗ ( M ⊗ M M ) → ( X ⊗ M ) ⊗ M M (3.9) or objects X ∈ Obj( C ) and M , M ∈ Obj( M ), and that all associativity and unit iso-morphisms are compatible in the sense that all triangle and pentagon diagrams commute.Let F : C → M , X X ⊗ M be the induction functor, which is in fact a tensor functor with functorial isomorphisms F ( X ⊗ X ) ∼ = −→ F ( X ) ⊗ F ( X )given by the composition( X ⊗ X ) ⊗ M A − X , X , M −−−−−−→ X ⊗ ( X ⊗ M ) Id X ⊗ l X ⊗ M −−−−−−−−→ X ⊗ ( M ⊗ ( X ⊗ M )) A X , M , X ⊗ M −−−−−−−−−→ ( X ⊗ M ) ⊗ ( X ⊗ M ) . Assume that F is a central functor, so that there is a braided tensor functor G : C → Z ( M )such that F = I ◦ G , where I : Z ( M ) → M is the forgetful functor.The following theorem is [DMNO, Lemma 3.5], but we add details and observe that neitherfiniteness nor semisimplicity is needed in the argument: Theorem 3.3.
In the setting of Theorem 3.2, assume that M is a tensor category, thenatural associativity isomorphisms (3.9) exist, and that the functor G M , M of the previoussubsection is representable. Assume in addition that C is a braided tensor category andinduction F : C → M is a central functor. Then the multiplication on A = Hom( M , M ) is commutative.Proof. We need to show that the multiplication map m ∈ Hom C ( A ⊗ A , A ) is commutative.Recalling that m = λ A ⊗ A (cid:0) λ − A (Id A ) ◦ λ − A (Id A ) ◦ A − A , A , M (cid:1) , we consider image of m under λ − A ⊗ A in Hom M ( F ( A ⊗ A ) , M ). By naturality of λ A ⊗ A wehave that λ − A ⊗ A ( m ◦ R A ⊗ A ) = λ − A ⊗ A ( m ) ◦ ( R A ⊗ A ⊗ Id M ) = λ − A ⊗ A ( m ) ◦ F ( R A ⊗ A ) , so we must show that λ − A ⊗ A ( m ) = λ − A ⊗ A ( m ) ◦ F ( R A ⊗ A ). To show this, we will use the diagram F ( A ⊗ A ) ∼ = / / F ( R A , A ) (cid:15) (cid:15) F ( A ) ⊗ F ( A ) λ − A (Id A ) ⊗ Id F ( A ) / / γ F ( A ) F ( A ) (cid:15) (cid:15) M ⊗ F ( A ) Id M ⊗ λ − A (Id A ) / / γ F ( A ) M (cid:15) (cid:15) M ⊗ M l M / / M F ( A ⊗ A ) ∼ = / / F ( A ) ⊗ F ( A ) Id F ( A ) ⊗ λ − A (Id A ) / / F ( A ) ⊗ M λ − A (Id A ) ⊗ Id M / / M ⊗ M r M : : tttttttttt The left square commutes because F lifts to the braided tensor functor G and because thebraiding isomorphism R ( F ( A ) ,γ F ( A ) ) , ( F ( A ) ,γ F ( A ) ) in Z ( M ) is given by γ F ( A ) F ( A ) . The square in themiddle commutes by the naturality of γ F ( A ) , and the pentagon commutes by Lemma 2.13and naturality of the unit isomorphisms.Because l M = r M we now see that it suffices to show that λ − A ⊗ A ( m ) is given by the top(equivalently the bottom) row of the diagram. Since by definition λ − A ⊗ A ( m ) = λ − A (Id A ) ◦ (Id A ⊗ λ − A (Id A )) ◦ A − A , A , M , his will follow from commutativity of the diagram F ( A ⊗ A ) A − A , A , M / / ∼ = ' ' PPPPPPPPPPPP A ⊗ F ( A ) Id A ⊗ λ − A (Id A ) / / F ( A ) λ − A (Id A ) / / M F ( A ) ⊗ F ( A ) Id F ( A ) ⊗ λ − A (Id A ) / / ∼ = O O F ( A ) ⊗ M λ − A (Id A ) ⊗ Id M / / r F ( A ) O O M ⊗ M r M O O In fact, the right square commutes by naturality of the unit isomorphisms, and recalling thedefinition of the functorial isomorphism F ( A ⊗ A ) → F ( A ) ⊗ F ( A ), we see that the trianglecommutes if the vertical isomorphism is (Id A ⊗ l A ⊗ M ) ◦ A − A , M , A ⊗ M . Then commutativityof the square in the middle follows from the commutative diagram( A ⊗ M ) ⊗ ( A ⊗ M ) A − A , M , A ⊗ M / / Id A ⊗ M ⊗ λ − A (Id A ) (cid:15) (cid:15) A ⊗ ( M ⊗ ( A ⊗ M )) Id A ⊗ l A ⊗ M / / Id A ⊗ (Id M ⊗ λ − A (Id A )) (cid:15) (cid:15) A ⊗ ( A ⊗ M ) Id A ⊗ λ − A (Id A ) (cid:15) (cid:15) ( A ⊗ M ) ⊗ M A − A , M , M / / A ⊗ ( M ⊗ M ) Id A ⊗ l M / / A ⊗ M together with the unit triangle constraint(Id A ⊗ l M ) ◦ A − A , M , M = (Id A ⊗ r M ) ◦ A − A , M , M = r A ⊗ M . This completes the proof that m ◦ R A , A = m . (cid:3) Canonical algebras.
We will now construct commutative algebras more concretely:we will see that what is called the canonical algebra associated to a suitable braided tensorcategory is always commutative.
Definition 3.4.
Let C be a monoidal category. The opposite category C op is the same as C as a category but has monoidal structure X ⊗ op Y := Y ⊗ X , associativity isomorphisms A op X , Y , Z = A − Z , Y , X , and unit isomorphisms l op X = r X , r op X = l X .By [EGNO, Example 7.4.2], a multitensor category C is a module category for the Deligneproduct C ⊠ C op with module map( X ⊠ Y ) ⊗ Z := ( X ⊗ Z ) ⊗ Y . If we assume in addition that C is braided, then C op also has a braiding given by R op X , Y = R − X , Y : X ⊗ op Y → Y ⊗ op X . Recalling the braid-reversed category C rev from Section 2.4, the identity functor on C givesa braided tensor equivalence between C rev and C op , with functorial isomorphisms R − Y , X : Id C ( X ⊗ Y ) → Id C ( X ) ⊗ op Id C ( Y ) . Thus we may view C as a module category for either C ⊠ C op or C ⊠ C rev . In this setting, thenatural associativity isomorphism A X ⊠ Y , Z , Z : ( X ⊠ Y ) ⊗ ( Z ⊗ Z ) → (( X ⊠ Y ) ⊗ Z ) ⊗ Z amounts to an isomorphism( X ⊗ ( Z ⊗ Z )) ⊗ Y → (( X ⊗ Z ) ⊗ Y ) ⊗ Z , hich is given by an appropriate combination of associativity isomorphisms together with R − Y , Z .To see that the induction functor F : C ⊠ C rev → C is a central functor, we note that F isnaturally isomorphic via the unit isomorphisms to the functor X ⊠ Y X ⊗ Y , f ⊠ g f ⊗ g. That is, F amounts to the extension to C ⊠ C rev of the identity functors from C and C rev into C , both of which are central functors lifting to Z ( C ) via X → ( X , R • , X ) and Y → ( Y , R − Y , • ),respectively. Since the images of these two functors in Z ( C ) centralize each other (recallM¨uger’s Proposition 2.15), the extension to C ⊠ C rev is also central. To be more specific, F lifts to the functor G : C ⊠ C rev → Z ( C ) given on objects by X ⊠ Y → ( X ⊗ Y , γ X ⊗ Y ) where γ X ⊗ YZ is given by the composition Z ⊗ ( X ⊗ Y ) A Z , X , Y −−−→ ( Z ⊗ X ) ⊗ Y R Z , X ⊗ Id Y −−−−−→ ( X ⊗ Z ) ⊗ Y A − X , Z , Y −−−→ X ⊗ ( Z ⊗ Y ) Id X ⊗R − Y , Z −−−−−→ X ⊗ ( Y ⊗ Z ) A X , Y , Z −−−→ ( X ⊗ Y ) ⊗ Z , as in (2.6).It now follows from Theorem 3.3 that if the functor G , : C ⊠ C rev → V ec given on objectsby X ⊠ Y Hom C (( X ⊗ ) ⊗ Y , ) ∼ = Hom C ( X ⊗ Y , )is representable, then A = Hom( , ) is a commutative associative algebra in C ⊠ C rev . Wehave two situations in which G , is representable. First, recalling Remark 3.1, this holdswhen C is a finite (in particular, rigid) braided tensor category. Secondly, when C is notnecessarily finite or rigid but is semisimple and has a contragredient functor, then G , isrepresentable if we replace C and C ⊠ C rev with their direct sum completions. In this case wecan take A = M X ∈ Irr( C ) X ′ ⊠ X where Irr( C ) is a set of equivalence class representatives for the simple objects in C . Notethat when C has infinitely many equivalence classes of simple objects, then A is not an objectof C ⊠ C rev but is an object of ( C ⊠ C rev ) fin ⊕ . For the braided tensor category structure onsuch completions, we refer again to [CGR] and especially [AR]; see also Appendix A.To describe the algebra structure on A more concretely, let us assume C is rigid and takesimple objects X , Y of C . We would like to determine the Z ∈ Irr( C ) for which m (( X ∗ ⊠ X ) ⊗ ( Y ∗ ⊠ Y )) ∩ ( Z ∗ ⊠ Z ) = 0. We first observe that by the definition of m and the naturality of λ , m | ( X ∗ ⊠ X ) ⊗ ( Y ∗ ⊠ Y ) = λ ( X ∗ ⊠ X ) ⊗ ( Y ∗ ⊠ Y ) (cid:0) λ − A ⊗ A ( m ) | (( X ∗ ⊠ X ) ⊗ ( Y ∗ ⊠ Y )) ⊗ (cid:1) . he morphism inside parentheses here is the right-side composition in the diagram(( X ∗ ⊠ X ) ⊗ ( Y ∗ ⊠ Y )) ⊗ A − X ∗ ⊠ X , Y ∗ ⊠ Y , (cid:15) (cid:15) ( i X ⊗ i Y ) ⊗ Id / / ( A ⊗ A ) ⊗ A − A , A , (cid:15) (cid:15) ( X ∗ ⊠ X ) ⊗ (( Y ∗ ⊠ Y ) ⊗ ) Id X ∗ ⊠ X ⊗ λ − Y ∗ ⊠ Y ( i Y ) (cid:15) (cid:15) i X ⊗ ( i Y ⊗ Id ) / / A ⊗ ( A ⊗ ) Id A ⊗ λ − A (Id A ) (cid:15) (cid:15) ( X ∗ ⊠ X ) ⊗ λ − X ∗ ⊠ X ( i X ) (cid:15) (cid:15) i X ⊗ Id / / A ⊗ λ − A (Id A ) s s ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ where i X and i Y represent the obvious inclusions. The diagram commutes by naturalityof associativity and λ . Moreover, because λ is an isomorphism, λ − X ∗ ⊠ X ( i X ) is a non-zeromorphism in Hom C (( X ∗ ⊗ ) ⊗ X , ) , a one-dimensional space spanned by d X : ( X ∗ ⊗ ) ⊗ X r X ∗ ⊗ Id X −−−−−→ X ∗ ⊗ X e X −→ , where e X is the evaluation morphism. Hence λ − X ∗ ⊠ X ( i X ) = a X d X for some a X = 0, and similarly λ − Y ∗ ⊠ Y ( i Y ) = a Y d Y for non-zero a Y .From this discussion, it follows that m | ( X ∗ ⊠ X ) ⊗ ( Y ∗ ⊠ Y ) = a X a Y λ ( X ∗ ⊠ X ) ⊗ ( Y ∗ ⊠ Y ) (cid:0) d X ◦ (Id X ∗ ⊠ X ⊗ d Y ) ◦ A − X ∗ ⊠ X , Y ∗ ⊠ Y , (cid:1) , and now the morphism inside parentheses is a morphism from(( X ∗ ⊠ X ) ⊗ ( Y ∗ ⊠ Y )) ⊗ = (( X ∗ ⊗ Y ∗ ) ⊠ ( X ⊗ op Y )) ⊗ = (( X ∗ ⊗ Y ∗ ) ⊗ ) ⊗ ( Y ⊗ X )to . In fact, we can identify X ∗ ⊗ Y ∗ = ( Y ⊗ X ) ∗ , and then this morphism is simply d Y ⊗ X = e Y ⊗ X ◦ ( r X ∗ ⊗ Y ∗ ⊗ Id Y ⊗ X ) (see for instance (see [BK, Tu, EGNO]). Now since C issemisimple, we have an isomorphism Y ⊗ X ∼ = L i ∈ I Z i where the Z i are simple objects of C and I is a finite index set. Under this isomorphism, d Y ⊗ X will be identified with the directsum of the d Z i which in turn will be identified with non-zero multiples of the inclusions Z ∗ i ⊗ Z i ֒ → A under λ Z i . Hence under the natural isomorphismHom C ⊠ C rev (( X ∗ ⊠ X ) ⊗ ( Y ∗ ⊠ Y ) , A ) ∼ = Y i,j ∈ I Hom C ⊠ C rev ( Z ∗ i ⊠ Z j , A ) ,m | ( X ∗ ⊠ X ) ⊗ ( Y ∗ ⊠ Y ) is sent to the product over i ∈ I of non-zero multiples of the inclusions of Z ∗ i ⊠ Z i into A . We conclude that Z ∗ ⊠ Z is included in m (( X ∗ ⊠ X ) ⊗ ( Y ∗ ⊠ Y )) preciselywhen Z occurs as a direct summand of Y ⊗ X (or equivalently X ⊗ Y since C is braided). Aslightly different explanation for this observation is given in [EGNO, Example 7.9.14].We have shown that if C is rigid, then the multiplication rules of A satisfy M Z ∗ ⊠ ZX ∗ ⊠ X , Y ∗ ⊠ Y = 1if and only if Z is a summand of X ⊗ Y . We can use this to show that A is a simple algebra:suppose I ֒ → A is a non-zero ideal of A . Because A is semisimple in ( C ⊠ C rev ) ⊕ and all simplesubobjects of A occur with multiplicity 1, any subobject such as I is also semisimple anda direct sum of certain X ∗ ⊠ X . For such X , we have X ∼ = X ∗∗ because C is braided (recall emark 2.2) and therefore M ⊠ X ∗∗ ⊠ X ∗ , X ∗ ⊠ X = 1. This means ⊠ ⊆ I , and then M Y ∗ ⊠ YY ∗ ⊠ Y , ⊠ = 1for any Y ∈ Irr( C ) implies I = A .We summarize the results of this section: Theorem 3.5.
Let C be a (not necessarily finite) semisimple braided tensor category with acontragredient functor. Then A = M X ∈ Irr( C ) X ′ ⊠ X is a commutative associative algebra in ( C ⊠ C rev ) fin ⊕ . If C is rigid, then A is simple and forsimple objects X , Y , Z of C , the multiplication rules are given by M Z ∗ ⊠ ZX ∗ ⊠ X , Y ∗ ⊠ Y ∗ = 1 if and onlyif Z is a summand of X ⊗ Y . Definition 3.6.
The algebra constructed in this subsection is called the canonical algebra in C ⊠ C rev (equivalently, in C ⊠ C op ). Remark 3.7.
Since commutative algebras are preserved by braided tensor equivalences, wecan restate Theorem 3.5 as follows. Let C be a semisimple braided tensor category with acontragredient functor, and suppose τ : C → D is a braid-reversed tensor equivalence (sothat τ : C rev → D is a braided equivalence). Then A = M X ∈ Irr( C ) X ′ ⊠ τ ( X )is a commutative associative algebra in ( C ⊠ D ) fin ⊕ , and if C is rigid, then A is simple.4. From algebras to braid-reversed equivalences
In the previous section, we showed how to construct a commutative associative algebrafrom a braid-reversed tensor equivalence. In this section, we consider the converse problem:given a simple algebra A in the Deligne product of two braided tensor categories, obtain abraid-reversed equivalence between the two factors of the Deligne product. Such a braid-reversed equivalence was obtained in [Lin] under the strong assumptions that the two braidedtensor categories are modular (in particular, finite) and that Rep A is semisimple. Here weobtain the equivalence without any finiteness assumptions and without any semisimplicityassumption on Rep A .4.1. Mirror equivalence.
In this section, we work in the following setting:(1) U is a (not necessarily finite) semisimple ribbon category, and { U i } i ∈ I is a subset ofdistinct simple objects in U that includes U . We use the notation U = U .(2) V is a ribbon category. In particular, both U and V are rigid.(3) We have a (commutative, associative, unital) algebra A = M i ∈ I M ii = M i ∈ I U i ⊠ V i . in C = U ⊠ V , or C ⊕ if I is infinite, where the V i are objects of V , not all assumed tobe simple, with V = V . Thus M = U ⊠ V = C , which we will denote by .
4) The tensor units U = U , V = V form a mutually commuting (or dual) pair in A ,in the sense thatdim Hom U ( U , U i ) = δ i, = dim Hom V ( V , V i ) . Note that the first equality is automatic because the U i are simple and distinct.(5) There is a partition I = I ⊔ I of the index set with 0 ∈ I such that for each i ∈ I j , j = 0 ,
1, the twist satisfies θ A | M ii = ( − j Id M ii . In particular, θ A = Id A .(6) Finally, A is simple as an object of Rep A .Note that although we are not assuming C = U ⊠ V is semisimple or finite, the conclusionof Proposition 2.6 still holds for the cases Y = A , A ∗ because we are assuming U is semisimpleand the U i are distinct. This means that we can use Corollary 2.11 together with the finalassumption on A to conclude that A ∗ is also simple as a right A -module (and in fact simplein Rep A since A is commutative).As a consequence of the assumption that U and V form a dual pair in A , we haveHom C ⊕ ( , A ) = M i ∈ I Hom C ( U ⊠ V , U i ⊠ V i )= M i ∈ I Hom U ( U , U i ) ⊗ F Hom V ( V , V i )= Hom C ( , U ⊠ V ) , which is the one-dimensional space F = End C ( ). This means A is what is called a haploid algebra in C (or C ⊕ ), and we may take ι A to be the canonical injection of = U ⊠ V intothe direct sum. We then define ε A : A → to be the canonical projection with respect tothe direct sum decomposition of A , so that ε A ◦ ι A = Id .Let U A ⊆ U and V A ⊆ V denote the full subcategories consisting of objects isomorphicto direct sums of objects appearing in the decomposition of A , that is, of the U i and V i ,respectively. Our main theorem will be a braid-reversed tensor equivalence between U A and V A , although we have not yet shown that they are tensor categories. The key idea is to usethe induction functor F : U ⊠ V →
Rep A to identify U A and V A with a common subcategoryof Rep A . More specifically, we will use the two tensor functors F U : U →
Rep AX
7→ F ( X ⊠ V ) f
7→ F ( f ⊠ Id V ) F V : V →
Rep AY
7→ F ( U ⊠ Y ) g
7→ F (Id U ⊠ g )First we show that F U and F V are fully faithful, so that U and V are tensor equivalent tosubcategories of Rep A : Lemma 4.1.
The functors F U and F V are fully faithful.Proof. We prove that F U is fully faithful; the proof for F V is the same (in particular, theproof does not use semisimplicity). We need to show that for any objects X , X in U , thelinear map F U : Hom U ( X , X ) → Hom
Rep A ( A ⊗ ( X ⊠ V ) , A ⊗ ( X ⊠ V ))is an isomorphism. We first observe that since in the Deligne product C = U ⊠ V we havethe isomorphismHom U ( X , X ) ⊗ F Hom V ( V , V ) ∼ = −→ Hom C ( X ⊠ V , X ⊠ V ) iven by f ⊗ F g f ⊠ g , and since Hom V ( V , V ) = F Id V , it is sufficient to show that F : Hom C ( X ⊠ V , X ⊠ V ) → Hom
Rep A ( A ⊗ ( X ⊠ V ) , A ⊗ ( X ⊗ V ))is an isomorphism.We show that F is an isomorphism by constructing an inverse: given F : A ⊗ ( X ⊠ V ) → A ⊗ ( X ⊗ V ) in Rep A , we define G ( F ) : X ⊠ V → X ⊠ V in C to be the composition X ⊠ V l − X ⊠ V −−−−→ ⊗ ( X ⊠ V ) ι A ⊗ Id X ⊠ V −−−−−−−→ A ⊗ ( X ⊠ V ) F −→ A ⊗ ( X ⊠ V ) ε A ⊗ Id X ⊠ V −−−−−−−→ ⊗ ( X ⊠ V ) l X ⊠ V −−−−→ X ⊠ V . For f ∈ Hom C ( X ⊠ V , X ⊠ V ), it is easy to see that G ( F ( f )) = f : in fact, G ( F ( f )) is thecomposition X ⊠ V l − X ⊠ V −−−−→ ⊗ ( X ⊠ V ) ( ε A ◦ ι A ) ⊗ f −−−−−−→ ⊗ ( X ⊠ V ) l X ⊠ V −−−−→ X ⊠ V . Using ε A ◦ ι A = Id and the naturality of the left unit isomorphisms, this reduces to f .On the other hand, given F : F ( X ⊗ V ) → F ( X ⊗ V ) in Rep A , F ( G ( F )) is thecomposition A ⊗ ( X ⊠ V ) Id A ⊗ l − X ⊠ V −−−−−−−→ A ⊗ ( ⊗ ( X ⊠ V )) Id A ⊗ ( ι A ⊗ Id X ⊠ V ) −−−−−−−−−−−→ A ⊗ ( A ⊗ ( X ⊠ V )) Id A ⊗ F −−−−→ A ⊗ ( A ⊗ ( X ⊠ V )) Id A ⊗ ( ε A ⊗ Id X ⊠ V ) −−−−−−−−−−−→ A ⊗ ( ⊗ ( X ⊠ V )) Id A ⊗ l X ⊠ V −−−−−−−→ A ⊗ ( X ⊠ V ) . (4.1)We first use the triangle axiom, the right unit property of A , and the naturality of theassociativity isomorphisms to rewriteId A ⊗ l X ⊠ V = ( r A ⊗ Id X ⊠ V ) ◦ A A , , X ⊠ V = ( µ A ⊠ Id X ⊠ V ) ◦ ((Id A ⊗ ι A ) ⊗ Id X ⊠ V ) ◦ A A , , X ⊠ V = ( µ A ⊠ Id X ⊠ V ) ◦ A A , A , X ⊠ V ◦ (Id A ⊗ ( ι A ⊗ Id X ⊠ V )) . Thus (4.1) becomes A ⊗ ( X ⊠ V ) Id A ⊗ e F −−−−→ A ⊗ ( A ⊗ ( X ⊠ V )) Id A ⊗ (( ι A ◦ ε A ) ⊗ Id X ⊠ V ) −−−−−−−−−−−−−→ A ⊗ ( A ⊗ ( X ⊠ V )) A A , A , X ⊠ V −−−−−−→ ( A ⊗ A ) ⊗ ( X ⊠ V ) µ A ⊗ Id X ⊠ V −−−−−−−→ A ⊗ ( X ⊠ V ) (4.2)where e F = F ◦ ( ι A ⊗ Id X ⊠ V )) ◦ l − X ⊠ V . Now we use the assumption that U and V form adual pair inside A to observe that e F ∈ Hom C ( X ⊠ V , A ⊗ ( X ⊠ V )) ∼ = Hom C X ⊠ V , M i ∈ I ( U i ⊠ V i ) ⊗ ( X ⊠ V ) ! ∼ = M i ∈ I Hom C ( X ⊠ V , ( U i ⊗ X ) ⊠ ( V i ⊗ V )) = M i ∈ I Hom U ( X , U i ⊗ X ) ⊗ F Hom V ( V , V i ) ∼ = Hom U ( X , U ⊗ X ) ⊗ F Hom V ( V , V ) ∼ = Hom C ( X ⊠ V , ⊗ ( X ⊠ V )) . In other words, the image of e F inside A ⊗ ( X ⊠ V ) is contained in ι A ( ) ⊗ ( X ⊠ V ). Since ι A ◦ ε A is the projection from A to ι A ( ) ⊆ A , it follows that (( ι A ◦ ε A ) ⊠ Id X ⊠ W ) ◦ e F = e F .Consequently, (4.2) becomes A ⊗ ( X ⊠ V ) Id A ⊗ l − X ⊠ V −−−−−−−→ A ⊗ ( ⊗ ( X ⊠ V )) Id A ⊗ ( ι A ⊗ Id X ⊠ V ) −−−−−−−−−−−→ A ⊗ ( A ⊗ ( X ⊠ V )) Id A ⊗ F −−−−→ A ⊗ ( A ⊗ ( X ⊠ V )) A A , A , X ⊠ V −−−−−−→ ( A ⊗ A ) ⊗ ( X ⊠ V ) µ A ⊗ Id X ⊠ V −−−−−−−→ A ⊗ ( X ⊠ V ) . Now because F is a morphism in Rep A and because µ F ( X i ⊠ V ) = ( µ A ⊗ Id X i ⊠ V ) ◦ A A , A , X i ⊠ V for i = 1 ,
2, we get A ⊗ ( X ⊠ V ) Id A ⊗ l − X ⊠ V −−−−−−−→ A ⊗ ( ⊗ ( X ⊠ V )) Id A ⊗ ( ι A ⊗ Id X ⊠ V ) −−−−−−−−−−−→ A ⊗ ( A ⊗ ( X ⊠ V )) A A , A , X ⊠ V −−−−−−→ ( A ⊗ A ) ⊗ ( X ⊠ V ) µ A ⊗ Id X ⊠ V −−−−−−−→ A ⊗ ( X ⊠ V ) F −→ A ⊗ ( X ⊠ V ) . Finally, the naturality of the associativity isomorphisms, the triangle axiom, and the rightunit property of µ A imply that this composition equals F , as required. (cid:3) Now the following result is key for showing that U A and V A are tensor subcategories and F U ( U A ) = F V ( V A ). It was proved in [Lin] under the assumption that U , V , and Rep A are allsemisimple; but even if U and V are modular tensor categories, Rep A is not guaranteed to besemisimple. By [KO, Theorem 3.3], Rep A is semisimple when A is a rigid algebra in C , whichby [KO, Lemma 1.20] means A is simple and dim C A = 0. Here we remove the assumptiondim C A = 0. Because the proof is lengthy and requires some preparatory lemmas, we willdefer it to Section 4.2. Key Lemma 4.2. If U i is simple and dim U U i = 0 , then F ( M i ) ∼ = F ( M ∗ i ) in Rep A . Remark 4.3.
In the setting of this section, with the U i simple objects in a semisimple tensorcategory, dim U U i = 0 is automatic by [EGNO, Proposition 4.8.4]. But as the proof of KeyLemma 4.2 does not use the semisimplicity of U , we have chosen to specify more preciselywhat conditions are needed for the result to hold.As a consequence of the Key Lemma, we will also prove the following in Section 4.2: Proposition 4.4.
The categories U A ⊆ U and V A ⊆ V are ribbon subcategories. Moreover, V A is semisimple with distinct simple objects { V i } i ∈ I . Key Lemma 4.2 and Proposition 4.4 already show that F U ( U A ) = F V ( V A ), so that byLemma 4.1, U A and V A are tensor equivalent. However, to show that U A and V A are braid-reversed equivalent, we will need to lift to the center. Let F U A and F V A denote the restrictionsof F U and F V to U A and V A , respectively, and let M = F U A ( U A ) ⊆ Rep A . By Key Lemma4.2 and Proposition 4.4, we also have M = F V A ( V A ). Then Lemma 4.1 and the discussion inSection 2.4 show that F U A and F V A are central functors that lift to braided tensor functors G U A : U A → Z ( M ) , G V A : V A → Z ( M ) , sing the braidings on U A and V A (see Section 2.4 for the precise definitions), as well as G rev U A : U rev A → Z ( M ) , G rev V A : V rev A → Z ( M )using the inverse braidings.With this setup, we can now conclude our main theorem: Theorem 4.5.
In the setting of this section, there is a braid-reversed tensor equivalence τ : U A → V A such that τ ( U i ) ∼ = V ∗ i for i ∈ I .Proof. By the universal property of Deligne products, the four fully faithful braided tensorfunctors G (rev) U A , G (rev) V A combine into braided tensor functors G U A ⊠ V A : U A ⊠ V A → Z ( M ) , G rev U A ⊠ V A : U rev A ⊠ V rev A → Z ( M ) . Now because U ⊠ V for U ∈ Obj( U A ) and U ⊠ V for V ∈ Obj( V A ) centralize each other in U A ⊠ V A (this follows from R U ⊠ V , U ⊠ V = R U , U ⊠ R V , V ), so do their images in Z ( M ) under G (rev) U A ⊠ V A . Thus G U A ( U A ) = G U A ⊠ V A ( U A ) ⊆ G U A ⊠ V A ( V A ) ′ = G V A ( V A ) ′ , G V A ( V A ) = G U A ⊠ V A ( V A ) ⊆ G U A ⊠ V A ( U A ) ′ = G U A ( U A ) ′ , G rev U A ( U rev A ) = G rev U A ⊠ V A ( U rev A ) ⊆ G rev U A ⊠ V A ( V rev A ) ′ = G rev V A ( V rev A ) ′ , G rev V A ( V rev A ) = G rev U A ⊠ V A ( V rev A ) ⊆ G rev U A ⊠ V A ( U rev A ) ′ = G rev U A ( U rev A ) ′ . It follows using M¨uger’s Proposition 2.15 that G U A ( U A ) ⊆ G V A ( V A ) ′ = G rev V A ( V rev A ) and G rev V A ( V rev A ) ⊆ G rev U A ( U rev A ) ′ = G U A ( U A ) , that is, G U A ( U A ) = G rev V A ( V rev A ). Hence we can get a braided tensor equivalence τ : U A → V rev A by composing G U A with an inverse to G rev V A . This is the same thing as a braid-reversed tensorequivalence τ : U A → V A , and τ ( U i ) ∼ = V ∗ i follows directly from Key Lemma 4.2. (cid:3) Proof of Key Lemma 4.2 and Proposition 4.4.
We use the same notation as inthe previous subsection. For future use, we shall prove Key Lemma 4.2 and Proposition 4.4in a slightly more general setting than that of the previous subsection: we allow A to be asuperalgebra in C ⊕ . More specifically, suppose that I = I ⊔ I is a partition of the indexset with 0 ∈ I , and set A = L i ∈ I M ii and A = L i ∈ I M ii ; assume that µ A is an evenmorphism, that is, µ A ( A i ⊗ A j ) ⊆ A i + j for i, j ∈ { , } , interpreting i + j mod 2 Z . Following [CKL], we say that A is:(1) An algebra of correct statistics if A is a commutative algebra and the twistsatisfies θ A = Id A (in this case we may assume I = ∅ ),(2) An algebra of wrong statistics if A is a commutative algebra and θ A = Id A ⊕ ( − Id A ),(3) A superalgebra of correct statistics if µ A | A j ⊗ A i ◦ R A i , A j = ( − ij µ A | A i ⊗ A j for i, j ∈ { , } and θ A = Id A ⊕ ( − Id A ), and(4) A superalgebra of wrong statistics if µ A | A j ⊗ A i ◦ R A i , A j = ( − ij µ A | A i ⊗ A j for i, j ∈{ , } and θ A = Id A . ote that both commutative algebras and superalgebras are monodromy-free: µ A ◦ R A , A = µ A . While in the previous subsection we only needed to assume A simple as an object ofRep A , here we will also assume A is simple as a right A -module. Lemma 4.6.
Assume A is simple both as a left and right A -module. Then there is aninvolution i i ′ of the index set I such that U i ′ ∼ = U ∗ i and V i ′ ∼ = V ∗ i . Moreover, there is aunique isomorphism ϕ i : M i ′ i ′ → M ∗ ii for each i ∈ I such that the diagram M i ′ i ′ ⊗ M iiϕ i ⊗ Id (cid:15) (cid:15) µ A / / A ε A (cid:15) (cid:15) M ∗ ii ⊗ M ii e M ii / / commutes, where e M ii = e U i ⊠ e V i is the coevaluation in C .Proof. The morphism ε A ◦ µ A : A ⊗ A → induces ϕ = Γ A , A ( ε A ◦ µ A ), the unique morphismin Hom C ( A , A ∗ ) making the diagram A ⊗ A ϕ ⊗ Id (cid:15) (cid:15) µ A / / A ε A (cid:15) (cid:15) A ∗ ⊗ A e A / / (4.3)commute. Since ε A ◦ µ A is non-zero by the unit property of A , ϕ is also non-zero. We willshow that ϕ is a homomorphism of right A -modules. Then since A is simple in Rep A , A ∗ is a simple right A -module by Corollary 2.11 and it will follow that ϕ is an isomorphism ofright A -modules.To show that ϕ is a right A -module homomorphism, we use the diagram( A ⊗ A ) ⊗ A ( ϕ ⊗ Id A ) ⊗ Id A u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ A − A , A , A (cid:15) (cid:15) µ A ⊗ Id A / / A ⊗ A µ A (cid:29) (cid:29) ❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀ ( A ∗ ⊗ A ) ⊗ A µ A ∗ ⊗ Id A (cid:15) (cid:15) A − A ∗ , A , A ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ A ⊗ ( A ⊗ A ) Id A ⊗ µ A / / ϕ ⊗ Id A ⊗ A (cid:15) (cid:15) A ⊗ A µ A & & ▼▼▼▼▼▼▼▼▼▼▼▼▼ ϕ ⊗ Id A (cid:15) (cid:15) A ∗ ⊗ A e A , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ A ∗ ⊗ ( A ⊗ A ) Id A ∗ ⊗ µ A / / A ∗ ⊗ A e A (cid:15) (cid:15) A ε A x x ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ which commutes as a consequence of the definitions of µ A ∗ and ϕ , the naturality of theassociativity isomorphisms, and the associativity of µ A . Using the outer compositions of thediagram and the naturality of Γ, we get e A ◦ (( µ A ∗ ◦ ( ϕ ⊗ Id A )) ⊗ Id A ) = Γ − A , A ( ϕ ) ◦ ( µ A ⊗ Id A ) = Γ − A ⊗ A , A ( ϕ ◦ µ A ) . Applying Γ A ⊗ A , A to both sides then yields the desired equality µ A ∗ ◦ ( ϕ ⊗ Id A ) = ϕ ◦ µ A .Now we have A ∗ = L i ∈ I M ∗ ii , and the factors M ∗ ii = U ∗ i ⊠ V ∗ i are inequivalent since the U i areinequivalent. Thus if we consider ϕ | M ii for any i , we see that the image must be contained n a unique M ∗ jj = U ∗ j ⊠ V ∗ j , and U ∗ j ∼ = U i . Thus if we set j = i ′ , we get the involution i i ′ of the index set I such that U i ′ ∼ = U ∗ i . Moreover, ϕ i := ϕ | M i ′ i ′ = ϕ | U i ′ ⊠ V i ′ : M i ′ i ′ = U i ′ ⊠ V i ′ → U ∗ i ⊠ V ∗ i = M ∗ ii must be an isomorphism. Since U i ′ is a simple object in U , we can identify ϕ i = f i ⊠ g i where f i : U i ′ → U ∗ i is a fixed (and unique up to scale) isomorphism and g i : V i ′ → V ∗ i isa morphism. But in fact g i must be an isomorphism as well because ϕ i is an isomorphism.Thus V i ′ ∼ = V ∗ i as well.Now we restrict the commutative diagram (4.3) to M i ′ i ′ ⊗ M ∗ ii = ( U i ′ ⊠ V i ′ ) ⊗ ( U i ⊠ V i ) ⊆ A ⊗ A . If we identify A ∗ ⊗ A = M i,j ∈ I ( U ∗ i ⊠ V ∗ i ) ⊗ ( U j ⊠ V j ) = M i,j ∈ I M ∗ ii ⊗ M jj , then under this identification, e A = X i ∈ I e M ii ◦ p i,i , where p i,j : A ⊗ A → M ∗ ii ⊗ M jj denotes the projection. Consequently, e A ◦ ( ϕ ⊗ Id A ) | M i ′ i ′ ⊗ M ii = X j ∈ I e M jj ◦ p j,j ◦ ( ϕ i ⊗ Id M ii ) = e M ii ◦ ( ϕ i ⊗ M ii ) , as desired. (cid:3) Remark 4.7.
Note that if A is a (super)algebra with I = ∅ (and 0 ∈ I ), Lemma 4.6together with the evenness of µ A imply that i ∈ I if and only if i ′ ∈ I .Let us use the notation e ij : M ∗ ij ⊗ M ij → to denote the evaluation e U i ⊠ e V j in C , andsimilarly for coevaluations. As usual in a rigid tensor category, we identify ∗ = withevaluation and coevaluation given by unit isomorphisms and their inverses (and similarly for U and V ). We will need the following simple lemma: Lemma 4.8.
For i, j ∈ I , the composition M ∗ ij ⊗ M ij ( l − U ∗ i ⊠ r − V ∗ j ) ⊗ ( r − U i ⊠ l − V j ) −−−−−−−−−−−−−→ ( M ∗ j ⊗ M ∗ i ) ⊗ ( M i ⊗ M j ) A M ∗ j ⊗ M ∗ i , M i , M j −−−−−−−−−−→ (( M ∗ j ⊗ M ∗ i ) ⊗ M i ) ⊗ M j A − M ∗ j , M ∗ i , M i ⊗ Id M j −−−−−−−−−−−→ ( M ∗ j ⊗ ( M ∗ i ⊗ M i )) ⊗ M j (Id M ∗ j ⊗ e i ) ⊗ Id M j −−−−−−−−−−−→ ( M ∗ j ⊗ ) ⊗ M j r M ∗ j ⊗ Id M j −−−−−−−→ M ∗ j ⊗ M j e j −→ is equal to e ij = e U i ⊠ e V j .Proof. This composition in U ⊠ V is the Deligne product of a morphism in U with a morphismin V . On the U side we get U ∗ i ⊗ U i l − U ∗ i ⊗ r − U i −−−−−→ ( U ⊗ U ∗ i ) ⊗ ( U i ⊗ U ) A U ⊗ U ∗ i , U i, U −−−−−−−−→ (( U ⊗ U ∗ i ) ⊗ U i ) ⊗ UA − U , U ∗ i , U i ⊗ Id U −−−−−−−−−→ ( U ⊗ ( U ∗ i ⊗ U i )) ⊗ U (Id U ⊗ e U i ) ⊗ Id U −−−−−−−−−−→ ( U ⊗ U ) ⊗ U U ⊗ Id U −−−−−−→ U ⊗ U l U = r U −−−−−→ U . By properties of unit isomorphisms, the first two arrows here equal r − U ⊗ U ∗ i ) ⊗ U i ◦ ( l − U ∗ i ⊗ Id U i ),and then we can use naturality to move this inverse right unit isomorphism to the end ofthe composition, where it cancels with r U . Thus we get U ∗ i ⊗ U i l − U ∗ i −−→ ( U ⊗ U ∗ i ) ⊗ U i A − U , U ∗ i , U i −−−−−→ U ⊗ ( U ∗ i ⊗ U i ) Id U ⊗ e U i −−−−−→ U ⊗ U r U = l U −−−−−→ U . Now the first two arrows are l − U ∗ i ⊗ U i , and naturality of the left unit isomorphisms implies thecomposition reduces to e U i , as required.The V side of the composition is similar: V ∗ j ⊗ V j r − V ∗ j ⊗ l − V j −−−−−→ ( V ∗ j ⊗ V ) ⊗ ( V ⊗ V j ) A V ∗ j ⊗ V , V , V j −−−−−−−−→ (( V ∗ j ⊗ V ) ⊗ V ) ⊗ V j A − V ∗ j , V , V ⊗ Id V j −−−−−−−−−→ ( V ∗ j ⊗ ( V ⊗ V )) ⊗ V j Id V ∗ j ⊗ ( l V = r V )) ⊗ Id V j −−−−−−−−−−−−−→ ( V ∗ j ⊗ V ) ⊗ V jr V ∗ j ⊗ Id V j −−−−−→ V ∗ j ⊗ V j e V j −−→ V . (4.4)We can use the triangle axiom to rewrite the first two arrows as ( r − V ∗ j ⊗ V ⊗ Id V j ) ◦ ( r − V ∗ j ⊗ Id V j ),and then the automorphism of V ∗ j resulting from the first five arrows is V ∗ j r − V ∗ j −−→ V ∗ j ⊗ V r − V ∗ j ⊗ V −−−−→ ( V ∗ j ⊗ V ) ⊗ VA − V ∗ j , V , V −−−−−−→ V ∗ j ⊗ ( V ⊗ V ) Id V ∗ j ⊗ r V −−−−−→ V ∗ j ⊗ V r V ∗ j −−→ V ∗ j . But the middle three arrows are Id V ∗ j ⊗ V by properties of the right unit isomorphisms, so thewhole composition is Id V ∗ j . Thus (4.4) is simply e V j . (cid:3) Now we can begin the proof of Key Lemma 4.2:
Proof.
Frobenius reciprocity and properties of duals show that we have natural isomorphismsHom
Rep A ( F ( M i ) , F ( M ∗ i )) ∼ = Hom C ( M i , A ⊗ M ∗ i ) ∼ = Hom C ( M i ⊗ M i , A ) ∼ = Hom C ( U i ⊠ V i , A ) , and similarly Hom Rep A ( F ( M ∗ i ) , F ( M i )) ∼ = Hom C ( U i ′ ⊠ V i ′ , A ) . Specifically, the inclusion U i ⊠ V i ֒ → A induces a Rep A -homomorphismΦ : F ( M i ) → F ( M ∗ i )given by the composition A ⊗ M i r − A ⊗ M i −−−−→ ( A ⊗ M i ) ⊗ Id A ⊗ i M i −−−−−→ ( A ⊗ M i ) ⊗ ( M i ⊗ M ∗ i ) assoc. −−−→ ( A ⊗ ( M i ⊗ M i )) ⊗ M ∗ i (Id A ⊗ ( r U i ⊠ l V i )) ⊗ Id M ∗ i −−−−−−−−−−−−−→ ( A ⊗ M ii ) ⊗ M ∗ i ( µ A | A ⊗ M ii ) ⊗ Id M ∗ i −−−−−−−−−−→ A ⊗ M ∗ i , here assoc. indicates the obvious composition of associativity isomorphisms in C . Similarly,the inclusion U i ′ ⊗ V i ′ ֒ → A induces a Rep A -morphismΨ : F ( M ∗ i ) → F ( M i )given by the composition A ⊗ M ∗ i r − A ⊗ M ∗ i −−−−→ ( A ⊗ M ∗ i ) ⊗ Id A ⊗ e i M ∗ i −−−−−→ ( A ⊗ M ∗ i ) ⊗ ( M ∗ i ⊗ M i ) assoc. −−−→ ( A ⊗ ( M ∗ i ⊗ M ∗ i )) ⊗ M i A ⊗ ϕ − i ◦ ( l U ∗ i ⊠ r V ∗ i )) ⊗ Id M i −−−−−−−−−−−−−−−−→ ( A ⊗ M i ′ i ′ ) ⊗ M i µ A | A ⊗ M i ′ i ′ ⊗ Id M i −−−−−−−−−−→ A ⊗ M i , where we have identified M i as the dual of M ∗ i using the coevaluation e i M ∗ i = R M i , M ∗ i ◦ ( θ M i ⊗ Id M ∗ i ) ◦ i M i ;the corresponding evaluation is e e M ∗ i = e M i ◦ R − M ∗ i , M i ◦ ( θ − M i ⊗ Id M ∗ i ) . We represent Φ and Ψ diagrammatically as follows:Φ = µ A A M i M i M ∗ i , Ψ = µ A θ A M ∗ i M i M ∗ i We will show that Φ ◦ Ψ = (dim U U i ) · Id F ( M ∗ i ) and Ψ ◦ Φ = ± (dim V V i ) · Id F ( M i ) , where the minus sign in the second equation occurs precisely when A is a (super)algebra ofwrong statistics and i ∈ I . This will mean that(dim U U i ) · Φ = Φ ◦ Ψ ◦ Φ = ± (dim V V i ) · Φ , and since Φ = 0 (it is induced by the non-zero inclusion U i ⊠ V i ֒ → A ), we will getdim U U i = ± (dim V V i ) . Since dim U U i = 0 by assumption, it will then follow that Φ is an isomorphism in Rep A withinverse (dim U U i ) − · Ψ. n order to calculate Φ ◦ Ψ, we first observe that by Frobenius reciprocity and propertiesof duals, we have natural isomorphismsHom
Rep A ( F ( M ∗ i ) , F ( M ∗ i )) ∼ = Hom C ( M ∗ i , A ⊗ M ∗ i ) ∼ = Hom C ( M ∗ i ⊗ M i , A ) . Under these identifications, a morphism F ∈ Hom
Rep A ( F ( M ∗ i ) , F ( M ∗ i )) corresponds to thecomposition M ∗ i ⊗ M i l − M ∗ i ⊗ Id M i −−−−−−→ ( ⊗ M ∗ i ) ⊗ M i ( ι A ⊗ Id M ∗ i ) ⊗ Id M i −−−−−−−−−−→ ( A ⊗ M ∗ i ) ⊗ M iF ⊗ Id M i −−−−−→ ( A ⊗ M ∗ i ) ⊗ M i A − A , M ∗ i, M i −−−−−−→ A ⊗ ( M ∗ i ⊗ C M i ) Id A ⊗ e M i −−−−−→ A ⊗ r A −→ A . (4.5)Thus to show Φ ◦ Ψ = (dim U U i ) · Id F ( M ∗ i ) , it follows from properties of the unit isomorphismsthat it suffices to show (4.5) with F = Φ ◦ Ψ reduces to (dim U U i ) · ( ι A ◦ e M i ).Similarly, we have a natural isomorphismHom Rep A ( F ( M i ) , F ( M i )) ∼ = Hom C ( M i ⊗ M ∗ i , A )under which G ∈ Hom
Rep A ( F ( M i ) , F ( M i )) corresponds to the composition M i ⊗ M ∗ i l − M i ⊗ Id M ∗ i −−−−−−→ ( ⊗ M i ) ⊗ M ∗ i ι A ⊗ Id M i ) ⊗ Id M ∗ i −−−−−−−−−−→ ( A ⊗ M i ) ⊗ M ∗ i G ⊗ Id M ∗ i −−−−−→ ( A ⊗ M i ) ⊗ M ∗ i A − A , M i , M ∗ i −−−−−−→ A ⊗ ( M i ⊗ M ∗ i ) Id A ⊗ e e M ∗ i −−−−−→ A ⊗ r A −→ A . (4.6)Again by properties of the unit isomorphisms, we need to show that the above compositionfor G = Ψ ◦ Φ reduces to ( ± dim V V i ) · ( ι A ◦ e e M ∗ i ).We now calculate (4.5) with F replaced by Φ ◦ Ψ, manipulating according to the followingtemplate: µ A µ A θ − M ∗ i M ∗ i M i M i M ∗ i M i A = µ A θ − M ∗ i M i M i M ∗ i = θ − M ∗ i M i M i M ∗ i θ − M ∗ i M i M i M ∗ i = (dim U U i ) M ∗ i M i . We start with the following map in Hom C ( M ∗ i ⊗ M i , A ), omitting subscripts from identitymorphisms to save space: M ∗ i ⊗ M i l − M ∗ i ⊗ Id −−−−→ ( ⊗ M ∗ i ) ⊗ M i ( ι A ⊗ Id) ⊗ Id −−−−−−→ ( A ⊗ M ∗ i ) ⊗ M ir − A ⊗ M ∗ i ⊗ Id −−−−−−→ (( A ⊗ M ∗ i ) ⊗ ) ⊗ M i (Id ⊗ e i M ∗ i ) ⊗ Id −−−−−−−→ (( A ⊗ M ∗ i ) ⊗ ( M ∗ i ⊗ M i )) ⊗ M iassoc. −−−→ (( A ⊗ ( M ∗ i ⊗ M ∗ i )) ⊗ M i ) ⊗ M i ((Id ⊗ ϕ − i ◦ ( l U ∗ i ⊠ r V ∗ i )) ⊗ Id) ⊗ Id −−−−−−−−−−−−−−−−−→ (( A ⊗ M i ′ i ′ ) ⊗ M i ) ⊗ M i ( µ A ⊗ Id) ⊗ Id −−−−−−→ ( A ⊗ M i ) ⊗ M i r − A ⊗ M i ⊗ Id −−−−−−→ (( A ⊗ M i ) ⊗ ) ⊗ M i (Id ⊗ i M i ) ⊗ Id −−−−−−−→ (( A ⊗ M i ) ⊗ ( M i ⊗ M ∗ i )) ⊗ M i assoc. −−−→ (( A ⊗ ( M i ⊗ M i )) ⊗ M ∗ i ) ⊗ M i ((Id ⊗ ( r U i ⊠ l V i )) ⊗ Id) ⊗ Id −−−−−−−−−−−−−→ (( A ⊗ M ii ) ⊗ M ∗ i ) ⊗ M i ( µ A ⊗ Id) ⊗ Id −−−−−−→ ( A ⊗ M ∗ i ) ⊗ M i A − A , M ∗ i, M i −−−−−−→ A ⊗ ( M ∗ i ⊗ M i ) Id ⊗ e M i −−−−→ A ⊗ r A −→ A . The first two simplifications to this composition come from the unit property of µ A and therigidity of M i . To achieve these simplifications, we first apply naturalities move ( ι A ⊗ Id) ◦ l − M ∗ i over several arrows in the composition before applying the unit property. We also applythe triangle axiom to r − A ⊗ M i and then naturality of associativity to collect all associativityisomorphisms from the latter half of the composition: M ∗ i ⊗ M i r − M ∗ i ⊗ Id −−−−→ ( M ∗ i ⊗ ) ⊗ M i (Id ⊗ e i M ∗ i ) ⊗ Id −−−−−−−→ ( M ∗ i ⊗ ( M ∗ i ⊗ M i )) ⊗ M i A M ∗ i, M ∗ i , M i ⊗ Id −−−−−−−−−→ (( M ∗ i ⊗ M ∗ i ) ⊗ M i ) ⊗ M i (( l − M ∗ i ⊗ Id) ⊗ Id) ⊗ Id −−−−−−−−−−→ ((( ⊗ M ∗ i ) ⊗ M ∗ i ) ⊗ M i ) ⊗ M i ( A , M ∗ i, M ∗ i ⊗ Id) ⊗ Id −−−−−−−−−−−→ (( ⊗ ( M ∗ i ⊗ M ∗ i )) ⊗ M i ) ⊗ M i ((Id ⊗ ϕ − i ◦ ( l U ∗ i ⊠ r V ∗ i )) ⊗ Id) ⊗ Id −−−−−−−−−−−−−−−−−→ (( ⊗ M i ′ i ′ ) ⊗ M i ) ⊗ M i ( l M i ′ i ′ ⊗ Id) ⊗ Id −−−−−−−−→ ( M i ′ i ′ ⊗ M i ) ⊗ M i Id ⊗ l − M i −−−−→ ( M i ′ i ′ ⊗ M i ) ⊗ ( ⊗ M i ) Id ⊗ ( i M i ⊗ Id) −−−−−−−→ ( M i ′ i ′ ⊗ M i ) ⊗ (( M i ⊗ M ∗ i ) ⊗ M i ) ssoc. −−−→ ( M i ′ i ′ ⊗ ( M i ⊗ M i )) ⊗ ( M ∗ i ⊗ M i ) µ A ◦ (Id ⊗ ( r U i ⊠ l V i )) ⊗ e M i −−−−−−−−−−−−−−→ A ⊗ r A −→ A . Now, the fourth and fifth arrows here are simply l − M ∗ i ⊗ M ∗ i , and then we can use naturalityto cancel l M i ′ i ′ with its inverse. Meanwhile, the pentagon axiom allows us to choose the firstisomorphism in the arrow marked assoc. to be (Id ⊗ Id) ⊗ A − M i , M ∗ i , M i , with the remain-ing associativity isomorphisms respecting the factor of M ∗ i ⊗ M i so that we can applyingnaturality of associativity to e M i : M ∗ i ⊗ M i r − M ∗ i ⊗ Id −−−−→ ( M ∗ i ⊗ ) ⊗ M i (Id ⊗ e i M ∗ i ) ⊗ Id −−−−−−−→ ( M ∗ i ⊗ ( M ∗ i ⊗ M i )) ⊗ M i A M ∗ i, M ∗ i , M i ⊗ Id −−−−−−−−−→ (( M ∗ i ⊗ M ∗ i ) ⊗ M i ) ⊗ M i ( ϕ − i ◦ ( l U ∗ i ⊠ r V ∗ i )) ⊗ Id) ⊗ Id −−−−−−−−−−−−−−→ ( M i ′ i ′ ⊗ M i ) ⊗ M i Id ⊗ l − M i −−−−→ ( M i ′ i ′ ⊗ M i ) ⊗ ( ⊗ M i ) Id ⊗ ( i M i ⊗ Id) −−−−−−−→ ( M i ′ i ′ ⊗ M i ) ⊗ (( M i ⊗ M ∗ i ) ⊗ M i ) Id ⊗A − M i, M ∗ i, M i −−−−−−−−−→ ( M i ′ i ′ ⊗ M i ) ⊗ ( M i ⊗ ( M ∗ i ⊗ M i )) Id ⊗ (Id ⊗ e M i ) −−−−−−−−→ ( M i ′ i ′ ⊗ M i ) ⊗ ( M i ⊗ ) assoc. −−−→ ( M i ′ i ′ ⊗ ( M i ⊗ M i )) ⊗ µ A ◦ (Id ⊗ ( r U i ⊠ l V i )) ⊗ Id −−−−−−−−−−−−→ A ⊗ r A −→ A . Now the rigidity of M i implies that the fifth through eigth arrows above collapse to Id ⊗ r − M i ,and further simplifications coming from properties of the right unit isomorphisms give: M ∗ i ⊗ M i r − M ∗ i ⊗ Id −−−−→ ( M ∗ i ⊗ ) ⊗ M i (Id ⊗ e i M ∗ i ) ⊗ Id −−−−−−−→ ( M ∗ i ⊗ ( M ∗ i ⊗ M i )) ⊗ M i A M ∗ i, M ∗ i , M i ⊗ Id −−−−−−−−−→ (( M ∗ i ⊗ M ∗ i ) ⊗ M i ) ⊗ M i ( ϕ − i ◦ ( l U ∗ i ⊠ r V ∗ i )) ⊗ Id) ⊗ Id −−−−−−−−−−−−−−→ ( M i ′ i ′ ⊗ M i ) ⊗ M i A − M i ′ i ′ , M i , M i −−−−−−−−→ M i ′ i ′ ⊗ ( M i ⊗ M i ) µ A ◦ (Id ⊗ ( r U i ⊠ l V i )) −−−−−−−−−−→ A . Now observe that the entire composition is a morphism inHom C ( M ∗ i ⊗ M i , A ) ∼ = Hom U ⊠ V ( U ⊠ ( V ∗ i ⊗ V i ) , A ) ∼ = M j ∈ I Hom U ( U , U j ) ⊗ Hom V ( V ∗ i ⊗ V i , V j ) . Since dim Hom U ( U , U j ) = δ ,j , it follows that the image must be contained in U ⊠ V = .Consequently, post-composing with ι A ◦ ε A has no effect on the composition, and we may useLemma 4.6 and naturality of the associativity isomorphisms to reduce to M ∗ i ⊗ M i r − M ∗ i ⊗ Id −−−−→ ( M ∗ i ⊗ ) ⊗ M i (Id ⊗ e i M ∗ i ) ⊗ Id −−−−−−−→ ( M ∗ i ⊗ ( M ∗ i ⊗ M i )) ⊗ M iassoc. −−−→ ( M ∗ i ⊗ M ∗ i ) ⊗ ( M i ⊗ M i ) ( l U ∗ i ⊠ r V ∗ i ) ⊗ ( r U i ⊗ l V i ) −−−−−−−−−−−−→ M ∗ ii ⊗ M ii e M ii −−→ ι A −→ A . Now using Lemma 4.8, we get M ∗ i ⊗ M i r − M ∗ i ⊗ Id −−−−→ ( M ∗ i ⊗ ) ⊗ M i (Id ⊗ e i M ∗ i ) ⊗ Id −−−−−−−→ ( M ∗ i ⊗ ( M ∗ i ⊗ M i )) ⊗ M i (Id ⊗ e M i ) ⊗ Id −−−−−−−−→ ( M ∗ i ⊗ ) ⊗ M i r M ∗ i ⊗ Id −−−−→ M ∗ i ⊗ M i e M i −−→ ι A −→ A . ince by definition e M i ◦ e i M ∗ i = dim C M i , (4.7)and since dim U U i = dim C M i , it follows that we get (dim U U i )( ι A ◦ e M i ), completing theproof that Φ ◦ Ψ = (dim U U i ) · Id F ( M ∗ i ) .Now we calculate Ψ ◦ Φ by considering equation (4.6) with G = Ψ ◦ Φ. We use the followingdiagrams as a guide: µ A µ A θ θ − A M i M i M ∗ i M i M ∗ i M ∗ i = µ A µ A A M i M i M ∗ i M ∗ i = µ A M i M i M ∗ i M ∗ i = µ A M i M i M ∗ i M ∗ i = ± µ A θ − M i M i M ∗ i M ∗ i = ± θ − M i M i M ∗ i M ∗ i = ± θ − θ − M i M ∗ i M i M ∗ i = ± (dim V V i ) θ − M i M ∗ i In this case, the relevant composition is: M i ⊗ M ∗ i l − M i ⊗ Id −−−−→ ( ⊗ M i ) ⊗ M ∗ i ι A ⊗ Id) ⊗ Id −−−−−−→ ( A ⊗ M i ) ⊗ M ∗ i r − A ⊗ M i ⊗ Id −−−−−−→ (( A ⊗ M i ) ⊗ ) ⊗ M ∗ i ⊗ i M i ) ⊗ Id −−−−−−−→ (( A ⊗ M i ) ⊗ ( M i ⊗ M ∗ i )) ⊗ M ∗ i assoc. −−−→ (( A ⊗ ( M i ⊗ M i )) ⊗ M ∗ i ) ⊗ M ∗ i ⊗ ( r U i ⊠ l V i )) ⊗ Id) ⊗ Id −−−−−−−−−−−−−→ (( A ⊗ M ii ) ⊗ M ∗ i ) ⊗ M ∗ i µ A ⊗ Id) ⊗ Id −−−−−−→ ( A ⊗ M ∗ i ) ⊗ M ∗ i r − A ⊗ M ∗ i ⊗ Id −−−−−−→ (( A ⊗ M ∗ i ) ⊗ ) ⊗ M ∗ i ⊗ e i M ∗ i ) ⊗ Id −−−−−−−→ (( A ⊗ M ∗ i ) ⊗ ( M ∗ i ⊗ M i )) ⊗ M ∗ i assoc. −−−→ (( A ⊗ ( M ∗ i ⊗ M ∗ i )) ⊗ M i ) ⊗ M ∗ i ⊗ ϕ − i ◦ ( r U ∗ i ⊠ l V ∗ i )) ⊗ Id) ⊗ Id −−−−−−−−−−−−−−−−−→ (( A ⊗ M i ′ i ′ ) ⊗ M i ) ⊗ M ∗ i µ A ⊗ Id) ⊗ Id −−−−−−→ ( A ⊗ M i ) ⊗ M ∗ i A − A , M i , M ∗ i −−−−−−→ A ⊗ ( M i ⊗ M ∗ i ) A ⊗ e e M ∗ i −−−−−→ A ⊗ r A −→ . As before, we can simplify using the unit property of A and the rigidity of M ∗ i (with evaluation e e M ∗ i and coevaluation e i M ∗ i ) to get: M i ⊗ M ∗ i r − M i ⊗ Id −−−−→ ( M i ⊗ ) ⊗ M ∗ i ⊗ i M i ) ⊗ Id −−−−−−−→ ( M i ⊗ ( M i ⊗ M ∗ i )) ⊗ M ∗ i A M i , M i, M ∗ i ⊗ Id −−−−−−−−−→ (( M i ⊗ M i ) ⊗ M ∗ i ) ⊗ M ∗ i r U i ⊠ l V i ) ⊗ Id) ⊗ Id −−−−−−−−−−→ ( M ii ⊗ M ∗ i ) ⊗ M ∗ i A − M ii, M ∗ i, M ∗ i −−−−−−−→ M ii ⊗ ( M ∗ i ⊗ M ∗ i ) Id ⊗ ϕ − i ◦ ( r U ∗ i ⊠ l V ∗ i ) −−−−−−−−−−→ M ii ⊗ M i ′ i ′ µ A −→ A . Also as before, we can post-compose with ι A ◦ ε A since dim Hom V ( V , V j ) = δ ,j .Now we use the (super)commutativity of µ A and properties of the twist θ A to replace µ A | M ii ⊗ M i ′ i ′ = ± µ A ◦ R − M i ′ i ′ , M ii ◦ (Id ⊗ θ − M i ′ i ′ ) , where the minus sign occurs precisely when A is a (super)algebra of wrong statistics and i ∈ I (recall Remark 4.7). Next applying naturality of R and θ to ϕ − i , and then usingLemma 4.6, our composition becomes, up to sign, M i ⊗ M ∗ i r − M i ⊗ Id −−−−→ ( M i ⊗ ) ⊗ M ∗ i ⊗ i M i ) ⊗ Id −−−−−−−→ ( M i ⊗ ( M i ⊗ M ∗ i )) ⊗ M ∗ i A M i , M i, M ∗ i ⊗ Id −−−−−−−−−→ (( M i ⊗ M i ) ⊗ M ∗ i ) ⊗ M ∗ i r U i ⊠ l V i ) ⊗ Id) ⊗ Id −−−−−−−−−−→ ( M ii ⊗ M ∗ i ) ⊗ M ∗ i A − M ii, M ∗ i, M ∗ i −−−−−−−→ M ii ⊗ ( M ∗ i ⊗ M ∗ i ) Id ⊗ ( r µ ∗ i ⊠ l V ∗ i ) −−−−−−−−→ M ii ⊗ M ∗ ii R − M ∗ ii, M ii ◦ (Id ⊗ θ M ∗ ii ) −−−−−−−−−−−→ M ∗ ii ⊗ M ii e M ii −−→ ι A −→ A . We now have the Deligne product of two morphisms in U and V , which we can calculateindividually. On the U side, we have U i ⊗ U ∗ i r − U i ⊗ Id −−−−→ ( U i ⊗ U ) ⊗ U ∗ i (Id ⊗ ( l − U = r − U )) ⊗ Id −−−−−−−−−−−→ ( U i ⊗ ( U ⊗ U )) ⊗ U ∗ i A U i, U , U ⊗ Id −−−−−−−−→ (( U i ⊗ U ) ⊗ U ) ⊗ U ∗ i ( r U i ⊗ Id) ⊗ Id −−−−−−−→ ( U i ⊗ U ) ⊗ U ∗ i A − U i, U , U ∗ i −−−−−→ U i ⊗ ( U ⊗ U ∗ i ) Id ⊗ l U ∗ i −−−−→ U i ⊗ U ∗ i R − U ∗ i , U i ◦ (Id ⊗ θ U ∗ i ) −−−−−−−−−→ U ∗ i ⊗ U i e U i −→ U ∼ −→ U . Now, the first six arrows collapse to the identity using the triangle axiom, and then we cancalculate the rest using properties of twists and duals: e U i ◦ R − U ∗ i , U i ◦ (Id ⊗ θ − U ∗ i ) = e U i ◦ ( θ − U ∗ i ⊗ Id) ◦ R − U ∗ i , U i = e U i ◦ (cid:0) ( θ − U i ) ∗ ⊗ Id (cid:1) ◦ R − U ∗ i , U i = e U i ◦ (Id ⊗ θ − U i ) ◦ R − U ∗ i , U i = e e U ∗ i . n the V side, we have the composition: V ⊗ V r − V ⊗ Id −−−−→ ( V ⊗ V ) ⊗ V (Id ⊗ i V i ) ⊗ Id −−−−−−−→ ( V ⊗ ( V i ⊗ V ∗ i )) ⊗ VA V , V i, V ∗ i ⊗ Id −−−−−−−−→ (( V ⊗ V i ) ⊗ V ∗ i ) ⊗ V ( l V i ⊗ Id) ⊗ Id −−−−−−−→ ( V i ⊗ V ∗ i ) ⊗ VA − V i, V ∗ i , V −−−−−→ V i ⊗ ( V ∗ i ⊗ V ) Id ⊗ r V ∗ i −−−−→ V i ⊗ V ∗ i R − V ∗ i , V i ◦ (Id ⊗ θ V ∗ i ) −−−−−−−−−→ V ∗ i ⊗ V i e V i −→ U ∼ −→ V . The third and fourth arrows here simplify to l V i ⊗ V ∗ i ⊗ Id, and then we can use naturality tocancel this left unit isomorphism with the first arrow of the composition. Moreover, the fifthand sixth arrows simplify to r V i ⊗ V ∗ i , and then we can use naturality to move this right unitisomorphism to the beginning of the composition: V ⊗ V r V = e V −−−−−→ V i V i −→ V i ⊗ V ∗ i R − V ∗ i , V i ◦ (Id ⊗ θ V ∗ i ) −−−−−−−−−→ V ∗ i ⊗ V i e V i −→ U ∼ −→ V . Now using the balancing equation, θ V = Id, and e V = e e V , we calculate e V i ◦ R − V ∗ i , V i ◦ (Id ⊗ θ − V ∗ i ) ◦ i V i ◦ e V = e V i ◦ θ − V ∗ i ⊗ V i ◦ R V i , V ∗ i ◦ ( θ V i ⊗ Id) ◦ i V i ◦ e e V = θ − V ◦ e V i ◦ e i V ∗ i ◦ e e V = (dim V V i ) e e V . In conclusion, we have shown that the composition in (4.6) with G = Ψ ◦ Φ equals ± (dim V V i )( ι A ◦ e e M i ) . This completes the proof that Φ and Ψ are both isomorphisms in Rep A . (cid:3) Finally we prove Proposition 4.4:
Proof.
Using Key Lemma 4.2, we have for i, j ∈ I the following isomorphisms in Rep A and/or C : F ( U i ⊠ V ∗ j ) ∼ = A ⊗ (( U i ⊠ V ) ⊗ ( U ⊠ V ∗ j )) ∼ = ( A ⊗ ( U i ⊠ V )) ⊗ ( U ⊠ V ∗ j ) ∼ = ( A ⊗ ( U ⊠ V ∗ i )) ⊗ ( U ⊠ V ∗ j ) ∼ = A ⊗ ( U ⊠ ( V ∗ i ⊗ V ∗ j )) ∼ = M k ∈ I U k ⊠ ( V k ⊗ ( V ∗ i ⊗ V ∗ j )) , (4.8)as well as F ( U i ⊠ V ∗ j ) ∼ = A ⊗ (( U ⊠ V ∗ j ) ⊗ ( U i ⊠ V )) ∼ = ( A ⊗ ( U ⊠ V ∗ j )) ⊗ ( U i ⊠ V ) ∼ = ( A ⊗ ( U j ⊠ V )) ⊗ ( U i ⊠ V ) ∼ = A ⊗ (( U j ⊗ U i ) ⊠ V ) ∼ = M k ∈ I ( U k ⊗ ( U j ⊗ U i )) ⊠ V k . (4.9)The first sequence of isomorphisms shows that F ( U i ⊠ V ∗ j ) is an object of ( U A ⊠ V ) ⊕ , andthen taking k = 0 in the second sequence of isomorphisms shows that F ( U i ⊠ V ∗ j ) contains( U j ⊗ U i ) ⊠ V as a subobject. This means ( U j ⊗ U i ) ⊠ V is an object of U A ⊠ V , that is, j ⊗ U i is an object of U A for all i, j ∈ I . This shows that U A is a tensor subcategory of U .Lemma 4.6 then shows that U A is closed under duals, and hence is a ribbon subcategory of U .Now we examine the summand of F ( U i ⊠ V ∗ j ) corresponding to U . On the one hand, (4.8)shows this is U ⊠ ( V ∗ i ⊗ V ∗ j ). On the other hand, (4.9) combined with semisimplicity of U implies it is isomorphic to U ⊠ L k ∈ I N k ∗ j,i V k , where N k ∗ j,i is the multiplicity of U ∗ k in U j ⊗ U i . Inother words, we have V ∗ i ⊗ V ∗ j ∼ = M k ∈ I N k ∗ j,i V k in V , or equivalently V j ⊗ V i ∼ = M k ∈ I N k ∗ j,i V k ′ . (4.10)This shows that V A is closed under tensor products (and also duals by Lemma 4.6) and thusis a ribbon subcategory of V .Now to show that V A is semisimple with the V i as distinct simple objects, we need to showthat dim Hom V ( V i , V j ) = δ i,j for i, j ∈ I . For this, we calculatedim Hom V ( V i , V j ) = dim Hom V ( V ⊗ V i , V j ) = dim Hom V ( V , V j ⊗ V i ′ )= X k ∈ I N k ∗ j,i ′ dim Hom V ( V , V k ′ ) = N j,i ′ = δ i,j , using properties of duals, equation (4.10), the fact that U and V form a dual pair in A ,and the simplicity and mutual inequivalence of the U i . (cid:3) Remark 4.9.
The above proof only shows that the V i are simple in V A , not that they arenecessarily simple in the possibly larger category V . For example, it is conceivable thatsome V i admits a non-trivial simple quotient V i / e V , provided that V i / e V does not occur as asubmodule of any other V j .5. From tensor categories to vertex operator algebras
Here we interpret the categorical theorems of the previous sections as theorems for vertexoperator algebras.5.1.
Vertex tensor categories.
Here to establish notation and terminology, we recall somefeatures and structures in the notion of vertex tensor category as formulated and developedin [HL1], [HL2]-[HL4], [Hu2], [HLZ1]-[HLZ8]; see also the exposition in [CKM, Section 3.1].In contrast to the preceding sections, here we will need to use the symbol ⊗ exclusively forvector space tensor products (over C ).We use the definitions of vertex operator algebra and module for a vertex operator algebrafrom [LL], except that we typically allow the Virasoro operator L (0) to act non-semisimplyon a module. Such modules are called grading-restricted generalized modules in[HLZ1]. To be more specific, a grading-restricted generalized module X has a C -grading X = L h ∈ C X [ h ] , where X [ h ] is the generalized L (0)-eigenspace with generalized eigenvalue h ,satisfying the two grading restriction conditions :(1) Each X [ h ] is finite-dimensional.(2) For any h ∈ C , X [ h − n ] = 0 for n ∈ N sufficiently large. he (vector space) tensor product of two vertex operator algebras U and V is a vertexoperator algebra [FHL], and if X and Y are grading-restricted, generalized U - and V -modules,respectively, then X ⊗ Y is a generalized U ⊗ V -module with( X ⊗ Y ) [ h ] = M h U + h V = h X [ h U ] ⊗ Y [ h V ] . The module X ⊗ Y will also satisfy the grading-restriction conditions if at least one of X and Y is strongly grading-restricted in the sense that there are finitely many cosets { h i + Z } in C / Z such that X [ h ] = 0 (or Y [ h ] = 0) only if h ∈ h i + Z for some i . From now on, we willrefer to such strongly grading-restricted, generalized modules simply as modules .A key feature of the tensor product theory of modules for a vertex operator algebra isa tensor product for each conformal equivalence class of spheres with two positively ori-ented punctures, one negatively oriented puncture, and local coordinates at the punctures.But to obtain braided tensor categories of vertex operator algebra modules, it is sufficient(see for instance [HLZ8]) to focus on P ( z )-tensor products, where P ( z ) is the sphere withpositively-oriented punctures at 0 and z ∈ C × , a negatively-oriented puncture at ∞ , andlocal coordinates w w , w w − z , w /w , respectively. Therefore, we shall here abuseterminology and use the term “vertex tensor category structure” to refer only to the tensorproduct functors and natural isomorphisms corresponding to the spheres P ( z ) for z ∈ C × .Given a vertex operator algebra V and a category C of V -modules, the P ( z )-tensor productof modules in C is defined in terms of P ( z )-intertwining maps. By [HLZ3, Proposition 4.8],these are precisely maps of the form Y ( · , z ) · , where Y : X ⊗ X → X [log x ] { x } is a (logarithmic) intertwining operator of type (cid:0) X X X (cid:1) for modules X , X , X in C and theformal variable x is specialized to z ∈ C × using a choice of branch of logarithm. The rangeof a P ( z )-intertwining map is the algebraic completion X , defined as the direct product(rather than direct sum) of the homogeneous graded subspaces of the module X .The P ( z )-tensor product of two modules X , X in C is defined to be a representing objectfor the functor V [ P ( z )] • X , X : C → V ec C , where for a module X in C , V [ P ( z )] XX , X is the spaceof P ( z )-intertwining maps of type (cid:0) XX X (cid:1) . That is, there are natural isomorphisms V [ P ( z )] XX , X ∼ −→ Hom C ( X ⊠ P ( z ) X , X ) . for all objects X in C . In particular, there is a distinguished P ( z )-intertwining map · ⊠ P ( z ) · of type (cid:0) X ⊠ P ( z ) X X X (cid:1) (corresponding to Id X ⊠ P ( z ) X ) such that if Y is any intertwining operatorof type (cid:0) VX X (cid:1) , there is a unique V -module homomorphism η Y : X ⊠ P ( z ) X → X such that η Y ( w ⊠ P ( z ) w ) = Y ( w , z ) w for w ∈ X , w ∈ X , where η Y is the natural extension of η Y to the algebraic completionsof X ⊠ P ( z ) X and X .In addition to the P ( z )-tensor product functors on the category C of V -modules, vertextensor category structure on C includes the following natural isomorphisms:(1) For continuous paths γ in C × beginning at z and ending at z , parallel transportisomorphisms T γ ; X , X : X ⊠ P ( z ) X → X ⊠ P ( z ) X .
2) For z ∈ C × , P ( z ) -unit isomorphisms l P ( z ) , X : V ⊠ P ( z ) X → X and r P ( z ); X : X ⊠ P ( z ) V → X .(3) For z , z ∈ C × satisfying | z | > | z | > | z − z | > P ( z , z ) -associativity iso-morphisms A P ( z ,z ); X , X , X : X ⊠ P ( z ) ( X ⊠ P ( z ) X ) → ( X ⊠ P ( z − z ) X ) ⊠ P ( z ) X . (4) For z ∈ C × , P ( z ) -braiding isomorphisms R P ( z ); X , X : X ⊠ P ( z ) X → X ⊠ P ( − z ) X . Remark 5.1.
The sphere P ( z , z ) in the associativity isomorphisms has three positivelyoriented punctures at 0, z , z and one negatively oriented puncture at ∞ . It can be obtainedeither by sewing together P ( z ) and P ( z ) spheres at the punctures 0 and ∞ , respectively,provided | z | > | z | , or by sewing together P ( z − z ) and P ( z ) spheres at the punctures ∞ and z , respectively, provided | z | > | z − z | . Thus the natural associativity isomorphisms ina vertex tensor category reflect the fact that these two sewing procedures yield conformallyequivalent spheres with punctures and local coordinates.For conditions on C guaranteeing the existence of these isomorphisms and details of theirconstruction, see [HLZ1]-[HLZ8]; see also the expository article [HL5] and [CKM, Section3.1]. In order to obtain braided tensor category structure from the vertex tensor categorystructure, one selects a particular tensor product functor, typically ⊠ P (1) which we shalldenote simply by ⊠ (it will be clear from the context when ⊠ denotes a Deligne productcategory and when ⊠ denotes a P (1)-tensor product). To obtain associativity and braidingisomorphisms for the single P (1)-tensor product, one needs to modify P (1)-braiding and P ( z , z )-associativity isomorphisms using parallel transport. For details, see [HLZ8].In general, it is difficult to show that a vertex tensor category C is rigid, but it willfrequently have a contragredient functor. Given a V -module X = L h ∈ C X [ h ] , the gradeddual vector space X ′ = L h ∈ C X ∗ [ h ] admits a V -module structure called the contragredientmodule [FHL]. If V is self-contragredient, that is, V ∼ = V ′ as a V -module, then X X ′ defines a contragredient functor. By [FHL, Proposition 5.3.2], X ′ is simple if and only if X is, and we have natural isomorphismsHom C ( X ⊠ Y , V ) ∼ = V [ P (1)] VX , Y ∼ = V [ P (1)] Y ′ X , V ′ ∼ = V [ P (1)] Y ′ X , V ∼ = Hom C ( X ⊠ V , Y ′ ) ∼ = Hom C ( X , Y ′ )given by the definition of the P (1)-tensor product, symmetries of intertwining operators(see for example [FHL, Proposition 5.5.2]), the isomorphism V ∼ = V ′ , and the right unitisomorphisms.The twist on a braided tensor category of modules is given by e πiL (0) . In particular, θ X isa scalar precisely when X is graded by a single coset of C / Z .5.2. Deligne products of vertex algebraic tensor categories.
In this section, we showthat under mild conditions, the Deligne product of braided tensor categories of modules fortwo vertex operator algebras is a braided tensor category of modules for the tensor productvertex operator algebra. Let U and V be vertex operator algebras, and let U and V bemodule categories for U and V , respectively, that admit vertex tensor category structure.We first consider when we can obtain vertex tensor category structure on a suitable categoryof U ⊗ V -modules. t is natural to consider the full subcategory C of U ⊗ V -modules whose objects are (iso-morphic to) direct sums of modules X ⊗ Y where X is a module in U and Y is a module in V . For the following theorem, we make fairly minimal assumptions on the vertex operatoralgebras U and V ; for similar results along these lines see for instance [Lin, Lemma 2.16] and[CKLinR, Proposition 3.3]. Theorem 5.2.
If all fusion rules among modules in either U or V are finite, then the category C of U ⊗ V -modules admits vertex tensor category structure as in [HLZ1] - [HLZ8] . Specifically,for modules U i in U and W , V i in V : (1) For z ∈ C × , P ( z ) -tensor products in C are given by ( U ⊗ V ) ⊠ P ( z ) ( U ⊗ V ) = ( U ⊠ P ( z ) U ) ⊗ ( V ⊠ P ( z ) V ) , with tensor product P ( z ) -intertwining map ⊠ P ( z ) = ⊠ P ( z ) ⊗ ⊠ P ( z ) . (2) For a continuous path γ in C × beginning at z and ending at z , the parallel transportisomorphism is T γ ; U ⊗ V , U ⊗ V = T γ ; U , U ⊗ T γ ; V , V . (3) For z ∈ C × , the P ( z ) -unit isomorphisms are l P ( z ); U i ⊗ V j = l P ( z ); U i ⊗ l P ( z ); V j and r P ( z ); U i ⊗ V j = r P ( z ); U i ⊗ r P ( z ); V j . (4) For z , z ∈ C × such that | z | > | z | > | z − z | > , the P ( z , z ) -associativityisomorphism is A P ( z ,z ); U ⊗ V , U ⊗ V , U ⊗ V = A P ( z ,z ); U , U , U ⊗ A P ( z ,z ); V , V , V . (5) For z ∈ C × , the P ( z ) -braiding isomorphism is R P ( z ); U ⊗ V , U ⊗ V = R P ( z ); U , U ⊗ R P ( z ); V , V . Moreover, if one of the categories U or V is semisimple and the other is closed under sub-modules and quotients, the category C is abelian and thus is a braided tensor category.Proof. Since the parallel transport and P ( z , z )-associativity isomorphisms in a vertex ten-sor category of modules for a vertex operator algebra are entirely characterized in terms oftensor product intertwining maps (see [HLZ8] for details), the indicated formulas for theseisomorphisms in C follow directly from the indicated identification of P ( z )-tensor prod-ucts and tensor product P ( z )-intertwining maps in C . The formulas for the P ( z )-unitisomorphisms and P ( z )-braiding isomorphisms also follow from these identifications, to-gether with the formulas Y U i ⊗ V j = Y U i ⊗ Y V j (from the definition in [FHL, Section 4.6]) and e zL U ⊗ V ( − = e zL U ( − ⊗ e zL V ( − (from the Leibniz formula). Moreover, all the isomorphismsindicated in the statement of the theorem are well defined; in particular, the convergenceof compositions of intertwining maps in C needed for the associativity isomorphisms followsfrom the convergence of intertwining maps in U and V . Moreover, all coherence proper-ties needed for a vertex tensor category follow from the corresponding coherence propertiessatisfied in U and V . o show that C admits vertex tensor category structure, then, it remains to show that formodules U , U in U and V , V in V , the pair (cid:0) ( U ⊠ P ( z ) U ) ⊗ ( V ⊠ P ( z ) V ) , ⊠ P ( z ) ⊗ ⊠ P ( z ) (cid:1) indeed satisfies the the universal property of a P ( z )-tensor product in C . For this, suppose X is any module in C and I is any P ( z )-intertwining map of type (cid:0) XU ⊗ V U ⊗ V (cid:1) . We mayidentify X with a (finite) direct sum X = L i U ( i ) ⊗ V ( i ) where the U ( i ) are modules in U and the V ( i ) are modules in V . Under this identification, [ADL, Theorem 2.10] implies thatthe intertwining map I may be identified with a (sum of) tensor products of intertwiningmaps: without loss of generality, we may assume that fusion rules in U are finite, so for any i , { I (1) i,j } J i j =1 is a basis for the space of P ( z )-intertwining maps of type (cid:0) U ( i ) U U (cid:1) . Then I = X i J i X j =1 I (1) i,j ⊗ I (2) i,j , where each I (2) i,j is a P ( z )-intertwining map of type (cid:0) V ( i ) V V (cid:1) .Now the universal property of P ( z )-tensor products in U and V imply that there areunique U -module homomorphisms η (1) i,j : U ⊠ P ( z ) U → U ( i ) such that I (1) i,j = η (1) i,j ◦ ⊠ P ( z ) , and there are unique V -module homomorphisms η (2) i,j : V ⊠ P ( z ) V → V ( i ) such that I (2) i,j = η (2) i,j ◦ ⊠ P ( z ) . Then the U ⊗ V -module homomorphism η = P i P J i j =1 η (1) i,j ⊗ η (2) i,j satisfies I = η ◦ ( ⊠ P ( z ) ⊗ ⊠ P ( z ) ).To show that η is the unique U ⊗ V -module homomorphism with this property, it suffices toshow that ⊠ P ( z ) ⊗ ⊠ P ( z ) is a surjective intertwining map in the sense that the U ⊗ V -module( U ⊠ P ( z ) U ) ⊗ ( V ⊠ P ( z ) V ) is generated by projections to the conformal weight spaces of theform π h (cid:0) ( u ⊠ P ( z ) u ) ⊗ ( w ⊠ P ( z ) w ) (cid:1) for h ∈ C , u ∈ U , u ∈ U , v ∈ V , and v ∈ V .In fact, π h (cid:0) ( u ⊠ P ( z ) u ) ⊗ ( v ⊠ P ( z ) v ) (cid:1) = X h U + h V = h π h U ( u ⊠ P ( z ) u ) ⊗ π h V ( v ⊠ P ( z ) v ) , where the sum is finite. Since the U ⊗ V -module generated by such projections is stable under L U (0) (and under L V (0)), this submodule contains each π h U ( u ⊠ P ( z ) u ) ⊗ π h V ( w ⊠ P ( z ) w )for h U , h V ∈ C . Such vectors span ( U ⊠ P ( z ) U ) ⊗ ( V ⊠ P ( z ) V ) by [HLZ3, Proposition 4.23],proving the desired the result. This completes the proof that C admits the indicated vertextensor category structure.Now to show that C is abelian, we may assume that U is semisimple and that V is closedunder submodules and quotients. Since C by definition includes direct sums, we just need toshow that every U ⊗ V -module homomorphism between modules in C has kernel and cokernelin C . For this, it is sufficient to show that C is closed under submodules and quotients.First, we note that by definition of C and semisimplicity of U , every module in C iscompletely reducible as a weak U -module, and every weak U -submodule of a module in C isalso completely reducible. Specifically, if X is a module in C , then X ∼ = L i U ( i ) ⊗ V ( i ) where he U ( i ) are distinct irreducible U -modules in U and the V ( i ) are V -modules in V . Then if e X ⊆ X is a weak U -submodule, e M ∼ = M i U ( i ) ⊗ e V ( i ) , where the e V ( i ) ⊆ V ( i ) are subspaces (possibly equal to zero). If e X is additionally an U ⊗ V -submodule, then the e V ( i ) are V -submodules of V ( i ) . Since V is closed under submodules, itfollows that the e V ( i ) are modules in V and e X is a module in C . Similarly, any quotient X / e X where X is a module in C and e X is an U ⊗ V -submodule is isomorphic to M / e M ∼ = M i U ( i ) ⊗ (cid:16) V ( i ) / e V ( i ) (cid:17) , where the U ( i ) are distinct irreducible U -modules in U , the V ( i ) are V -modules in V , and the e V ( i ) are V -submodules. Since V is closed under quotients, it follows that X / e X is a module in C . (cid:3) Remark 5.3.
Because the braided tensor category structure on C derives from the vertextensor category structure, we have the following identifications of structure isomorphisms inthe braided tensor category structure:(1) The unit isomorphisms are l U i ⊗ V j = l U i ⊗ l V j and r U i ⊗ V j = r U i ⊗ r V j . (2) The associativity isomorphisms are A U ⊗ V , U ⊗ V , U ⊗ V = A U , U , U ⊗ A V , V , V . (3) The braiding isomorphisms are R U ⊗ V , U ⊗ V = R U , U ⊗ R V , V . Remark 5.4.
The simple identifications of structure isomorphisms in C with tensor prod-ucts of structure isomorphisms in U and V do not follow simply from the existence of anisomorphism ( U ⊗ V ) ⊠ P ( z ) ( U ⊗ V ) ∼ = ( U ⊠ P ( z ) U ) ⊗ ( V ⊠ P ( z ) V ) , but also from the identification ⊠ P ( z ) = ⊠ P ( z ) ⊗ ⊠ P ( z ) under this isomorphism.Now we can show that C is actually the Deligne product of U and V : Theorem 5.5.
Suppose U and V are locally finite abelian categories, one of U and V issemisimple, and the other is closed under submodules and quotients. Then C is braidedtensor equivalent to the Deligne product category U ⊠ V . roof. The functor
U × V → C given by ( X , Y ) X ⊗ Y on objects and ( f, g ) f ⊗ g onmorphisms is right exact in both variables, so there is a unique functor F : U ⊠ V → C determined on objects by F ( X ⊠ Y ) = X ⊗ Y and F ( f ⊠ g ) = f ⊗ g on morphisms.Conversely, we may assume that U is semisimple and define a functor G : C → U ⊠ V on objects by G ( X ⊗ Y ) = X ⊠ Y for a simple object X in U and any object Y in V . Formorphisms, we observe that if X and X are simple in U , then every morphism inHom C ( X ⊗ Y , X ⊗ Y )can be written as f ⊗ g where f : X → X is fixed (and is either an isomorphism or 0)and g : Y → Y is some V -module homomorphism. Thus we can define G on morphisms by G ( f ⊗ g ) = f ⊠ g for such f and g .Now, F ◦ G is naturally isomorphic to Id C because it is an additive functor that equalsthe identity on indecomposable objects of C . For the other direction, the functor G ◦ F ◦ ⊠ : U × V → U ⊠ V is right exact in both variables and sends ( X , Y ) to X ⊠ Y when X is a simplemodule in U . Since the universal property of U ⊠ V implies that the identity is the onlyendofunctor of U ⊠ V with this property (up to natural isomorphism), we have G ◦F ∼ = Id U ⊠ V .Now because U and V are locally finite, spaces of intertwining operators are finite di-mensional, so Theorem 5.2 applies showing C is an (abelian) braided tensor category. ThenRemark 5.3 shows that F is compatible with the braided tensor category structures on U ⊠ V and C , and thus is an equivalence of braided tensor categories. (cid:3) Algebras in vertex tensor categories.
The foundational theorem for algebras inbraided tensor categories of modules for a vertex operator algebra is the following result ofHuang, Kirillov, and Lepowsky:
Theorem 5.6. [HKL, Theorem 3.2, Remark 3.3]
Let C be a category of modules for a vertexoperator algebra V that admits vertex tensor category structure as in [HLZ1] - [HLZ8] and thusalso braided tensor category structure. Then the following two notions are equivalent: (1) A vertex operator algebra ( A , Y A , , ω ) in C (with the same vacuum and conformalvectors as V ). (2) A commutative associative algebra ( A , µ A , ι A ) in C with injective unit and trivial twist: θ A = Id A . Since we are also concerned with algebras in C ⊕ , or more particularly algebras in C fin ⊕ when C is semisimple, we need the generalization of this theorem to such algebras. We note thatthe conformal weight gradings of objects A = L s ∈ S A s will not necessarily satisfy gradingrestriction conditions, but such an object can still be a conformal vertex algebra in the senseof [HLZ1]. Theorem 5.7.
Let C be a semisimple category of modules for a vertex operator algebra V that admits vertex tensor category structure as in [HLZ1] - [HLZ8] and thus also braided tensorcategory structure. Then the following two notions are equivalent: (1) A conformal vertex algebra ( A , Y A , , ω ) in C fin ⊕ with the same vacuum and conformalvectors as V . (2) A commutative associative algebra ( A , µ A , ι A ) in C fin ⊕ with injective unit and θ A = Id A . roof. Since the proof is the same as that of [HKL, Theorem 3.2] with minor changes, weonly indicate how to obtain a vertex operator Y A from an algebra multiplication µ A , and viceversa.Given a commutative associative algebra A = L s ∈ S A s with injective unit, trivial twist,and multiplication map µ A = { ( µ A ) s ,s ,t } s ,s ,t ∈ S ∈ Y ( s ,s ,t ) ∈ S × S × S Hom C ( A s ⊠ A s , A t ) , the vertex operator Y A is defined to be unique the intertwining operator Y A of (weak) V -modules that satisfies Y A ( a , a = X t ∈ S ( µ A ) s ,s ,t ( a ⊠ a )for a ∈ A s , a ∈ A s . The sum is well defined because µ A is a morphism in C ⊕ and thus forfixed s , s ∈ S , ( µ A ) s ,s ,t = 0 for all but finitely many t ∈ S .Conversely, given a conformal vertex algebra A = L s ∈ S A s in C fin ⊕ with the same vacuumand conformal vector as V , we would like to define µ A ∈ Hom C ⊕ ( A ⊠ A , A ) ⊆ Y ( s ,s ,t ) ∈ S × S × S Hom C ( A s ⊠ A s , A t )to be the tuple { ( µ A ) s ,s ,t } s ,s ,t ∈ S where( µ A ) s ,s ,t : A s ⊠ A s → A t is the unique morphism such that( µ A ) s ,s ,t ( a ⊠ a ) = π t ( Y A ( a , a )for a ∈ A s , a ∈ A s , where π t is the canonical projection from A to A t . However, we needto show that for fixed s , s ∈ S , we have ( µ A ) s ,s ,t = 0 for all but finitely many t ∈ T . Infact, this holds because A is an object of C fin ⊕ . For, if π t ◦ Y A | A s ⊗ A s = 0, it is a non-zerointertwining operator of type (cid:0) A t A s A s (cid:1) , and then Hom C ( A s ⊠ A s , A t ) = 0. But since A s ⊠ A s is a direct sum of finitely many simple objects in C and because these finitely many simpleobjects can occur in only finitely many A t , this space of homomorphisms is non-zero for onlyfinitely many t . (cid:3) Remark 5.8.
We may replace the condition θ A = Id A with θ A = Id A if we wish to allow Z -graded conformal vertex algebra extensions of V . Remark 5.9.
The conclusion of Theorem 5.7 also applies when C = U ⊠ V where U issemisimple and V is not, provided we restrict our attention to algebras of the form A = M i ∈ I U i ⊗ V i where { U i } i ∈ I is a set of simple modules in U containing any given simple module of U finitely many times. .4. The main theorems for vertex operator algebras.
We can now combine Theorem3.5, Remark 3.7, Proposition 4.4, Theorem 4.5, Theorem 5.5, Theorem 5.6, and Theorem5.7 into the following fundamental theorem relating conformal vertex algebra extensions oftensor product vertex operator algebras to braid-reversed equivalences:
Theorem 5.10.
Let U and V be locally finite module categories for simple self-contragredientvertex operator algebras U and V , respectively, that are closed under contragredients andadmit vertex tensor category structure as in [HLZ1] - [HLZ8] and thus also braided tensorcategory structure. Assume moreover that U is semisimple and V is closed under submodulesand quotients. (1) Suppose { U i } i ∈ I is a set of representatives of equivalence classes of simple modulesin U with U = U and τ : U → V is a braid-reversed tensor equivalence with twistssatisfying θ τ ( U i ) = ± τ ( θ − U i ) for i ∈ I . Then A = M i ∈ I U ′ i ⊗ τ ( U i ) is a Z -graded conformal vertex algebra extension of U ⊗ V . Moreover, if U is rigid,then A is simple and the multiplication rules of A satisfy M U ′ k ⊗ τ ( U k ) U ′ i ⊗ τ ( U i ) , U ′ j ⊗ τ ( U j ) = 1 if andonly if U k occurs as a submodule of U i ⊠ U j . (2) Conversely, suppose U and V are both ribbon categories, { U i } i ∈ I is a set of distinctsimple modules in U with U = U , and A = M i ∈ I U i ⊗ V i is a simple Z -graded conformal vertex algebra extension of U ⊗ V , where the V i areobjects of V satisfying dim Hom V ( V , V i ) = δ i, and there is a partition I = I ⊔ I of the index set with ∈ I and M i ∈ I j U i ⊗ V i = M n ∈ j + Z A ( n ) for j = 0 , . Let U A ⊆ U , respectively V A ⊆ V , be the full subcategories whose objectsare isomorphic to direct sums of the U i , respectively of the V i . Then: (a) U A and V A are ribbon subcategories of U and V respectively. Moreover, V A issemisimple with distinct simple objects { V i } i ∈ I . (b) There is a braid-reversed equivalence τ : U A → V A such that τ ( U i ) ∼ = V ′ i for all i ∈ I . Remark 5.11.
Note that for part (2) of the theorem, we have dim Hom U ( U , U i ) = δ i, anddim U U i = 0 for i ∈ I because the U i are simple objects in a semisimple ribbon category.As discussed in the Introduction, part (1) of Theorem 5.10 provides a partial answer toa question of Chongying Dong, while part (2) allows us to address a general question onthe rationality of coset extensions of the form U ⊗ V ⊆ A : If U and V are strongly rationalvertex operator algebras (that is, simple, self-contragredient, CFT-type, C -cofinite, and ational), is the extension A also strongly rational? In particular, is the category of grading-restricted, generalized A -modules semisimple? We answer these questions using results from[KO] together with Theorem 5.10 and [ENO, Theorem 2.3]: Theorem 5.12.
Suppose U and V are braided fusion categories of modules for simple self-contragredient vertex operator algebras U and V , respectively, and A = M i ∈ I U i ⊗ V i is a simple Z -graded vertex operator algebra extension of U ⊗ V in C = U ⊠ V where the U i are distinct simple modules in U including U = U and the V i are modules in V such that dim Hom V ( V , V i ) = δ i, . Then dim C A > and the category of (grading-restricted, generalized) A -modules in C is abraided fusion category.Proof. The rigidity and semisimplicity of the braided tensor category Rep A of A -modulesin C (and indeed of the larger tensor category Rep A ) follow from [KO, Theorem 1.15] and[KO, Theorems 3.2 and 3.3] provided dim C A = 0. Then to see why Rep A has finitelymany isomorphism classes of simple modules, let { M j } Jj =1 be a set of equivalence class rep-resentatives of simple modules in C . Because C is semisimple, any irreducible module X inRep A contains at least one such M j , and the U ⊗ V -module inclusion M j ֒ → X togetherwith Frobenius reciprocity imply there is a non-zero A -module homomorphism F J M j =1 M j ! → X , which is a surjection because X is simple. Thus every irreducible A -module in C is a quotientof F (cid:16)L Jj =1 M j (cid:17) , and it suffices to show that this module in Rep A has finitely many distinctirreducible quotients. Since Rep A is semisimple, it suffices to show that F (cid:16)L Jj =1 M j (cid:17) isfinitely generated. In fact, since the M j are simple modules in C , each F ( M j ) = A ⊠ M j issingly-generated as an A -module by any non-zero m j ∈ M j .It remains to show that dim C A >
0. The braid-reversed tensor equivalence guaranteedby part (2) of Theorem 5.10 impliesdim C A = X i ∈ I (dim U U i )(dim V V i ) = X i ∈ I (dim U U i )(dim U rev U ′ i ) . We note that since we assume A is Z -graded, each U i ⊗ V i must be Z -graded, which meansthat the proper twist to use for calculating dimensions in U rev is θ − .Now for each i ∈ I , recall the isomorphism δ U i : U i → U ′′ i of Remark 2.2. By [ENO,Theorem 2.3], we have Tr U i ( δ U i )Tr U ′ i (( δ − U i ) ′ ) > , where Tr U i ( δ U i ), for instance, is defined by the composition U i U i −→ U i ⊠ U ′ i δ U i ⊠ Id U ′ i −−−−−→ U ′′ i ⊠ U ′ i e U ′ i −→ U . he definition of δ U i shows that Tr U i ( δ U i ) = dim U U i , so we just need to show thatTr U ′ i (( δ − U i ) ′ ) = dim U rev U ′ i . We use the definitions of Tr U ′ i (( δ − U i ) ′ ) and dual of a homomorphism to obtainTr U ′ i (( δ − U i ) ′ ) = e U ′′ i ◦ (( δ − U i ) ′ ⊠ Id U ′′ i ) ◦ i U ′ i = e U i ◦ (Id U ′ i ⊠ δ − U i ) ◦ i U ′ i . On the other hand, using the definition of δ U i we have e U ′ i = e U i ◦ R U i , U ′ i ◦ ( θ U i ⊠ Id U ′ i ) ◦ ( δ − U i ⊠ Id U ′ i )= e U i ◦ (Id U ′ i ⊠ δ − U i ) ◦ R U ′′ i , U ′ i ◦ ( θ U ′′ i ⊠ Id U ′ i ) , so that e U i ◦ (Id U ′ i ⊠ δ − U i ) ◦ i U ′ i = e U ′ i ◦ ( θ − U ′′ i ⊠ Id U ′ i ) ◦ R − U ′′ i , U ′ i ◦ i U ′ i = e U ′ i ◦ (( θ − U ′ i ) ′ ⊠ Id U ′ i ) ◦ R − U ′′ i , U ′ i ◦ i U ′ i = e U ′ i ◦ (Id U ′′ i ⊠ θ − U ′ i ) ◦ R − U ′′ i , U ′ i ◦ i U ′ i = e U ′ i ◦ R − U ′′ i , U ′ i ◦ ( θ − U ′ i ⊠ Id U ′′ i ) ◦ i U ′ i = dim U rev U ′ i , as required. (cid:3) If the vertex operator algebras U and V of the above theorem are strongly rational, we cannow show that A will also be strongly rational, provided it is CFT-type. Self-contragrediencyand C -cofiniteness follow from the corresponding properties of U and V via [Li1, Theorem3.1] and [ABD, Proposition 5.2]. Moreover, the argument in Lemma 3.6 and Proposition 3.7of [CM] (see also [McR, Proposition 4.15]) shows that A is rational provided the categoryof grading-restricted generalized A -modules is semsimple, which is the content of Theorem5.12. Thus we have: Corollary 5.13.
In the setting of Theorem 5.12, suppose U and V are strongly rationalvertex operator algebras. If A is simple and CFT-type, then A is strongly rational. Appendix A. Direct sum completion
In this Appendix, we gather the main constructions from [AR] of the direct sum comple-tion of a category. Given a category C with possibly additional structures, C ⊕ is essentiallythe smallest category closed under arbitrary direct sums. One may restrict to only countabledirect sums, and this would be enough for our purposes. Even if C is abelian, one cannotguarantee that C ⊕ is abelian. Hence, we may wish to consider the smallest category contain-ing C closed under direct sums, kernels and cokernels; see [CGR]. If C is already semisimple,then C ⊕ is also abelian (see for example Section 3.5 of [Ja]).If C is a braided tensor category, C ⊕ can be also turned into a braided tensor category.If C has a system of isomorphisms θ X that satisfy balancing, we get a system of balancingisomorphisms in C ⊕ as well. However, even if C is rigid, we cannot guarantee rigidity of C ⊕ .But we shall not need C ⊕ to be rigid. et C be a C -linear additive category. We define the direct sum completion C ⊕ as follows.The objects of C ⊕ are:Obj( C ⊕ ) = (M s ∈ S X s (cid:12)(cid:12) S is a set , X s ∈ Obj( C ) for all s ∈ S ) . (A.1)The morphisms are:Hom C ⊕ M s ∈ S X s , M t ∈ T Y t ! = (cid:8)(cid:0) α, { f s,t } s ∈ S,t ∈ α ( s ) (cid:1)(cid:9) / ∼ , (A.2)with the following definitions.(1) Let P fin ( S ) denote the set of finite subsets of a set S . α : P fin ( S ) → P fin ( T ) is afunction that commutes with unions. By abuse of notation, we write α ( s ) = α ( { s } )for all s ∈ S . Since it is enough to specify α on singletons, and we will often do so.Sometimes, α will map singletons to singletons, in which case, we shall simply write α ( s ) = t (or α : s t ) if α ( { s } ) = { t } and { f s } s ∈ S in place of { f s } s ∈ S,t ∈ T .(2) f s,t ∈ Hom C ( X s , Y t ) for all s ∈ S, t ∈ α ( s ).(3) ∼ is an equivalence relation defined by: (cid:0) α, { f s,t } s ∈ S,t ∈ α ( s ) (cid:1) ∼ (cid:0) β, { g s,t } s ∈ S,t ∈ β ( s ) (cid:1) (A.3)iff all of the following are satisfied:(a) f s,t = 0 if t ∈ α ( s ) \ β ( s ),(b) f s,t = g s,t if t ∈ α ( s ) ∩ β ( s ),(c) g s,t = 0 if t ∈ β ( s ) \ α ( s ).The identity morphism on L s ∈ S X s is given by (cid:0) Id P fin ( S ) , { Id X s } s ∈ S (cid:1) . Note that we can alsocharacterize morphism spaces as follows:Hom C ⊕ M s ∈ S X s , M t ∈ T Y t ! ⊆ Y s ∈ S,t ∈ T Hom C ( X s , Y t )is the subset of tuples ( f s,t ) s ∈ S,t ∈ T such that for any fixed s ∈ S , f s,t = 0 for all but finitelymany t ∈ T .There are natural candidates for C -vector space structure on the morphism spaces, for azero object, zero morphisms, and direct sums. With these, it was shown in [AR] that C ⊕ isin fact again a C -linear additive category. There is also a fully faithful functor I : C → C ⊕ as follows: X M i ∈{ } X i with X := X f (cid:0) Id { } , { f s,t = f } s ∈{ } ,t ∈{ } (cid:1) . We will sometimes abuse the notation and write X = I ( X ).If C is a tensor category, the tensor product bifunctor on C ⊕ is defined by: M s ∈ S X s ⊗ M t ∈ T Y t = M ( s,t ) ∈ S × T X s ⊗ Y t , (A.4) α, { f s,s ′ } s ∈ S,s ′ ∈ α ( s ) (cid:1) ⊗ (cid:0) β, { g t,t ′ } t ∈ T,t ′ ∈ β ( t ) (cid:1) = (cid:0) α × β, { f s,s ′ ⊗ g t,t ′ } ( s,t ) ∈ S × T, ( s ′ ,t ′ ) ∈ α ( s ) × β ( t ) (cid:1) . (A.5)The unit object of C ⊕ is I ( C ) = L s ∈{ } X s with X = . The structure morphisms aredefined as follows. Let X = M s ∈ S X s , Y = M t ∈ T Y t , Z = M u ∈ U Z u . (A.6)If C is rigid, let X ∗ = M s ∈ S X ∗ s . (A.7)Then define l X = ( α : (0 , s ) s, { f ,s = l X s } s ∈ S ) : ⊗ X → X , (A.8) r X = ( α : ( s, s, { f s, = r X s } s ∈ S ) : X ⊗ → X , (A.9) A X , Y , Z = ( α : ( s, ( t, u )) (( s, t ) , u ) , { f ( s, ( t,u )) = A X s , Y t , Z u } ( s, ( t,u )) ∈ S × ( T × U ) ): X ⊗ ( Y ⊗ Z ) → ( X ⊗ Y ) ⊗ Z , (A.10) R X , Y = ( α : ( s, t ) ( t, s ) , { f ( s,t ) = R X s , Y t } ( s,t ) ∈ S × T ) : X ⊗ Y → Y ⊗ X , (A.11) e X = ( α : ( s ′ , s ) δ s ′ = s , { f s ′ ,s = δ s ′ = s e X s } ( s ′ ,s ) ∈ S × S : X ∗ ⊗ X → , (A.12) θ X = ( α : s s, { f s = θ X s } s ∈ S ) : X → X . (A.13)These definitions give requisite structures on C ⊕ , except for rigidity. In particular, θ X isproved to satisfy the balancing axiom in [AR]. References [ABD] T. Abe, G. Buhl and C. Dong, Rationality, regularity, and C -cofiniteness, Trans. Amer. Math. Soc. (2004), 3391–3403.[ADL] T. Abe, C. Dong and H. Li, Fusion rules for the vertex operator algebra M (1) and V + L , Comm. Math. Phys. (2005), 171–219.[AFO] M. 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Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton,Alberta, Canada.
E-mail: [email protected]
Department of Mathematics, University of Denver, Denver, Colorado, USA.
E-mail: [email protected]
Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA.
E-mail: [email protected]@vanderbilt.edu