Computing fusion rules for spherical G-extensions of fusion categories
aa r X i v : . [ m a t h . QA ] S e p COMPUTING FUSION RULES FOR SPHERICAL G -EXTENSIONS OFFUSION CATEGORIES MARCEL BISCHOFF AND COREY JONES
Abstract. A G -graded extension of a fusion category C yields a categorical action ρ : G → Aut br ⊗ ( Z ( C )). If the extension admits a spherical structure, we provide a method for recov-ering the fusion rules in terms of the action ρ . We then apply this to find closed formulas forthe fusion rules of extensions of some group theoretical categories and of cyclic permutationcrossed extensions of modular categories. Contents
1. Introduction 11.1. Acknowledgments 42. Preliminaries 42.1. Modular categories 52.2. Equivariantization 72.3. Extension theory 93. Recovering fusion rules from the Lagrangian algebra 103.1. The canonical Lagrangian algebra 123.2. G -extensions 153.3. Example: Fusion categories Morita equivalent to Vec( A ) 173.4. Example: Fusion categories with center tensor equivalent to Vec( B ) 194. Examples from G -crossed extensions of modular categories 204.1. Cyclic permutation actions 24References 27Appendix A. Fusion rules for Z / Z permutation extensions of Fib 301. Introduction
The theory of fusion categories has found significant applications in the study of 2Dquantum physics, most notably in conformal field theory [MS90, FRS02, BKLR15, HL13] andtopological phases of matter [Kit06, NSS +
08, Wan10]. In both these contexts modular tensorcategories appears as important invariants of physical models. If the model has a group G ofglobal symmetries, one obtains a G -crossed braided fusion category which is a G -extension ofthe original invariant [Kir02, M¨ug05, BBCW14]. This makes understanding of G -extensions Date : September 9, 2019.M.B. is supported by NSF Grant DMS-1700192/1821162.C.J. is supported by NSF Grant DMS-1901082. f fusion categories of fundamental importance for the study of two dimensional symmetryenriched physical systems.The theory developed by Etingof, Nikshych, and Ostrik [ENO10] provides the basic tools toconstruct and classify G -extensions of fusion categories. They show that every extension of C can be constructed from a braided categorical action ρ : G → Aut br ⊗ ( Z ( C )), provided a certaincohomology class o ( ρ ) ∈ H ( G, C × ) is trivial. In the case this obstruction vanishes, thepossible extensions associated with this action form a torsor over the group H ( G, C × ), butall the extensions have the same fusion rules. This tells us that in principle the fusion rulesof the extension can be computed from the initial categorical action ρ . However in practicethis problem is usually difficult. Naively following the proofs of the above statements from[ENO10] requires the computation of the data for the associated bimodule categories, theirrelative Deligne products, and the bimodule functors used to define the monoidal producton the extension (see Section 2.3). The amount of computation required to find the data inthese intermediate steps quickly becomes infeasible as the rank of the fusion category grows.In this paper, we provide a method for computing the fusion rules of an extension inan elementary way from a detailed knowledge of ρ and Z ( C ). Our approach bypasses thecomputation of the associated bimodule categories and their data. It allows for the derivationof closed form expressions of fusion rules for families of extensions in some general cases. Thekey observation in our approach is that the fusion rules can be recovered from the compositionand convolution operations on the space of endomorphisms of the canonical Lagrangianalgebra I ( ) ∈ Z ( C ) (see Corollary 3.3). This may be viewed as a direct generalization ofcharacter theory for the representation category of a finite group .We now give an outline of how this works. It is well known that End( I ( )) ∼ = K ( C ) ⊗ Z C as associative complex algebras. However, given End( I ( )) as an abstract algebra, to findthe fusion rules we need more information. We also need to identify the canonical basiselements { [ X ] } X ∈ Irr( C ) (or perhaps some scaled version of them) so that we can recover thefusion rules by examining the coefficients under multiplication.Luckily there is an additional operation on End( I ( )) that allows us to recover the (appro-priately scaled) canonical basis in a canonical way. For any commutative Frobenius algebra A in a braided fusion category, there are two associative binary operations on the vectorspace End( A ). The first is the usual composition of morphisms, which in general is noncom-mutative. The second is the convolution operation ∗ (see equation 4, [Bis17, BD18]). Thisoperation makes the vector space End( A ) into a commutative algebra in the usual sense. If A = I ( ) ∈ Z ( C ) is the canonical Lagrangian algebra, we show the minimal idempotents e X with respect to ∗ are in bijective correspondece with equivalence classes simple objects X ∈ Irr( C ) (see equation 7). Since the algebra (End( I ( ) , ∗ ) is commutative and semi-simple, the minimal idempotents give a canonical basis for the space End( I ( )). We thenshow that e X ◦ e Y = P Z ∈ Irr( C ) d X d Y d Z N ZXY e Z (where d X indicates (any) spherical dimensionfunction, see Proposition 3.2). Thus while the basis { e X } X ∈ Irr( C ) is not quite the canonicalbasis { [ X ] } X ∈ Irr( C ) for the fusion ring described above, the quantity d X d Y d Z is independent ofthe spherical structure, and thus we can recover the fusion rules N ZXY by examining thenumbers C ZXY defined by ( e X ◦ e Y ) ∗ e Z = C ZXY e Z , and renormalizing (see equation 8).Therefore, for a G -extension C ⊆ D , the fusion rules of D can be determined by computingthe composition and convolution products on the endomorphisms of the canonical Lagrangianalgebra in Z ( D ). By [GNN09], the latter category is equivalent to the equivariantization of he G -crossed braided relative center Z C ( D ). Here the G action restricts to ρ on the triviallygraded component Z ( C ) ⊆ Z D ( C ). Furthermore, the canonical Lagrangian algebra for D lives in the subcategory Z ( C ) G ⊆ Z ( D ). In particular, if I G : Z ( C ) → Z ( C ) G is the adjoint ofthe forgetful functor F G : Z ( C ) G → Z ( C ), then I D ( ) ∼ = I G ( I C ( )) ∈ Z ( C ) G ⊆ Z ( D ). Usingthe adjunction between I G and F G , we can compute the triple (End( I D ) , ∗ , ◦ ) terms of thedata of Z ( C ) and the category action ρ . (Section 3.2). We can then recover the fusion rulesas described above.A subtlety is that for the numbers we produce to actually be the fusion rules of theextension, we need to assume that D admits a spherical structure, though we do not need toexplicitly choose one (see Remark 3.4). Unfortunately, the extension theory of [ENO10] hasnot been developed to take spherical structures into account, hence it is not clear a-prioriif the G -extensions constructed from a given categorical action admit spherical structures.However, if we make the mild assumption that C is pseudo-unitary , then any extension isautomatically so and hence our hypothesis is satisfied (see Proposition 2.6). In this case ourresults apply to any G -extension, without additional hypothesis.As a first application, we utilize our method to give general formulas for fusion rules of G -extenisons of Vec( ˆ A × A, q ) L where A is an abelian group, q is the canonical hyperbolicquadratic form, and L ≤ ˆ A × A is a Lagrangian subgroup. The extensions depend onan initial braided categorical action on Vec( ˆ A × A, q ) ∼ = Z (Vec( ˆ A × A, q ) L ) (see Theorem3.6 ). Here Vec( ˆ A × A, q ) L denotes the fusion category of modules of the group algebra objectassociated to the Lagrangian subgroup. Note that the categories Vec( ˆ A × A, q ) L are preciselythose which are Morita equivalent to Vec( A ).We then focus on the case when C is modular, and the categorical action can be factored G → Aut br ⊗ ( C ) → Aut br ⊗ ( Z ( C )), where the second functor acts on the right factor in Z ( C ) ∼ = C ⊠ C rev . If a corresponding extension exists, it has the additional structure of a G -crossedbraided extension of C . These are the extensions which naturally appear both in conformalfield theory [M¨ug05] and topological phases of matter [BBCW14], hence are of the greatestinterest in applications. In this case, the nice form of the Lagrangian algebra and of theaction allows us to describe the convolution product in a general way.The examples of this type we consider are permutation actions on C ⊠ n . These have longbeen of interest to physicists in the context of rational conformal field theory [BHS98, Ban02,LX04, KLX05, M¨ug05] as an intermediate step in the study of permutation orbifold theories.More recently, permutation extensions have been of interest in the theory of topologicalphases under the guise of “genons” for their potential in quantum computing applications[BJQ13, BBCW14].Permutation crossed extensions have also come to attract the attention of mathematicians.They have been studied from the point of view of modular functors [BS11]. From an algebraicviewpoint, the o obstruction for permutation actions was shown to vanish in [GJ19], hencethese extensions always exist. They have been studied in the Z / Z case ([BS11, BFRS10],[EMJP18, Pas18]). Very recently, Delaney has given an algorithm for computing the fusionrules of general permutation extensions using the concept of bare defects [Del19]. Here wewill use our method to give a closed formula for the fusion rules in the case of maximal cyclic permutation extensions (see Theorem 4.5 ). Our formulas for the fusion rules involve thedimensions of vectors spaces assigned by the modular functor derived from C to surfaces ith field insertions. While our approach for fusion rules is different from [Del19], we haveverified that their algorithm produces the same numbers as our formula in several examples.The outline of the paper is as follows. The preliminary section briefly collects some factsabout fusion categories, modular categories, equivariantizations, and extension theory thatwill be used in the paper. In Section 3, we demonstrate how to reconstruct the basedfusion ring from the canonical Lagrangian algebra in Z ( C ) and apply this to G -extensions asdescribed above. Finally, we turn to the case of G -crossed extensions of modular categories,giving explicit examples of the computation of fusion rules for G -extensions from a givencategorical action. We an include an appendix with a list of fusion rules for the Z / Z cyclicpermutation extension of the modular category Fib ⊠ Acknowledgments.
The authors would like to thank Colleen Delaney, Cain Edie-Michell, Dave Penneys and Julia Plavnik for very useful discussions and comments on anearly draft. We also thank Colleen Delaney for sharing an early draft of [Del19] with usand for coordinating arXiv submissions. Marcel Bischoff was supported by NSF grant DMS-1700192/1821162. Corey Jones was supported by NSF Grant DMS-1901082.2.
Preliminaries
Recall a fusion category is a linear, finitely semi-simple, rigid, monoidal category withsimple unit [EGNO15, Chapter 4]. The semi-simplicity gives us well behaved fusion rules ,described by the non-negative integers N ZXY = dim( C ( X ⊗ Y, Z )) for
X, Y, Z ∈ Irr( C ). Hereand for the rest of the paper, we use Irr( C ) to denote a fixed choice of representative for eachequivalence classes of simple object in C . If C is any category, we use here and throughout thepaper the notation C ( X, Y ) := Hom(
X, Y ). We typically use f ◦ g to represent compositionof morphisms.For fusion categories, there are several notions of dimension that are important to con-sider. First, there is a unique function FPdim : Irr( C ) → R + such that FPdim( ) = 1and FPdim( X ) FPdim( Y ) = P Z ∈ Irr( C ) N ZXY
FPdim( Z ) called the Frobenius-Perron dimen-sion , [EGNO15, Section 3.3]. This dimension depends only on the based ring K ( C ), and isinsensitive to the details of the categorification.The second notion of dimension depends on a choice of spherical structure . This is amonoidal natural isomorphism from the identity to the double dual functor X ¯¯ X suchthat the associated left and right pivotal traces are equal [EGNO15, Chapter 4.7]. Aspherical structure gives us a single, well-defined spherical trace for every object X ∈ C ,Tr X : C ( X, X ) → C . We can then define the spherical dimension function d : Irr( C ) → R =0 , d X := Tr X (1 X ) which satisfies d X d Y = P Z ∈ Irr( C ) N ZXY d Z . Spherical structures also allow us tomake use of the spherical graphical calculus, which we use freely. [BW99, Tur94, BK01, Sel11].It is an open question whether every fusion category admits a spherical structure [ENO05].There is a third important notion of dimension in fusion categories. Let X be a simple ob-ject in a fusion category, and let ¯ X be a (two-sided) dual object. Choose arbitrary evaluationand coevaluation morphisms R X ∈ C ( , ¯ X ⊗ X ) , ¯ R X ∈ C ( , X ⊗ ¯ X ) , R ∗ X ∈ C ( ¯ X ⊗ X, ) , ¯ R ∗ X ∈C ( , X ⊗ ¯ X ) satisfying the duality equations. Then the quantity d { X, ¯ X } := ( R ∗ X ◦ R X )( ¯ R ∗ X ◦ ¯ R X ) (1) s called the paired dimension and does not depend on the choices of ¯ X or evaluation andcoevaluation morphisms. This number is strictly positive [Bar16], and thus we can definethe fusion dimension of X as the positive square root d + X := q d { X, ¯ X } . We note that we are using different notation from [Bar16] for the various dimensions. Inparticular we use d + X for the fusion dimension instead of for the Frobenius-Perron dimension.Like the Frobenius-Perron dimension, the paired dimension and the fusion dimension areintrinsic to a fusion category and do not depend on a choice of additional structure. However,the fusion dimension depends on the associator of the category and cannot be determinedby the fusion ring alone. Definition 2.1.
A fusion category is pseudo-unitary if FPdim ( X ) = d { X, ¯ X } for all X ∈ Irr( C ) . We note that our definition has many equivalent formulations (see [ENO05, Section 8.3,8.4]. If C is pseudo-unitary, there exists a canonical spherical structure on C whose sphericaldimensions are the Frobenius-Perron dimensions [ENO05, Proposition 8.23]. All unitaryfusion categories are pseudo-unitary, and thus in applications (most relevant) to physics allexamples are pseudo-unitary.2.1. Modular categories.
Recall a braided fusion category is a fusion category equippedwith a family of natural isomorphisms σ X,Y : X ⊗ Y → Y ⊗ X satisfying a family of coherences(namely the hexagon axioms). If { X ∈ Irr( C ) : σ Y,X ◦ σ X,Y = 1 X ⊗ Y for all Y ∈ Irr( C ) } = { } , then we say C is non-degenerately braided , or simply non-degenerate. If C is non-degenerate and in addition equipped with a spherical structure, we say C is modular .We refer the reader to [BK01] for an overview of modular categories, modular data, andsome their important properties. Here we use the conventions S X,Y := Tr X, ¯ Y ( σ ¯ Y ,X ◦ σ X, ¯ Y ).Furthermore, we use dim C := P X ∈ Irr( C ) d X which is a positive number independent of thespherical structure. We use √ dim C to denote the positive square root. Non-degeneracy ofthe category C is equivalent to the invertibility of the matrix S [M¨ug02].For modular categories we have the relation S = dim( C ) · C where C X,Y = δ X, ¯ Y is the charge conjugation matrix, hence S − X,Y = C ) S X, ¯ Y .In applications to both high energy and low energy physics, the fusion categories whichappear typically are naturally modular. Modular categories are also significant as they(essentially) classify 3-2-1 topological quantum field theories [BDSPV15]. In particular,given a modular tensor category one assigns a vector space to a genus g surface with markedpoints labelled by objects of C (which we will call “field insertions”) [BK01]. The dimensionof this vector space is also the dimension of the n -point conformal blocks on a genus g surfaceassociated with the matrix S of C when C arises as the modular tensor category associated toa completely rational conformal field theory [MS89]. A formula for this dimension is stated n [MS89, Eq. (A.7)]. With our normalization of S it reads as M C ( g, X ; · · · , X n ) := X Y ∈ Irr( C ) S X ,Y S ,Y · · · S X n ,Y S ,Y √ dim C S ,Y ! g − (2)= (dim C ) g − X Y ∈ Irr( C ) S X ,Y · · · S X n ,Y d n +2 g − Y . Let us write N Y ,...,Y m X ,...,X n = dim C ( X ⊠ · · · ⊠ X n , Y ⊠ · · · ⊠ Y m ) . For g ∈ Z ≥ we define the genus g fusion coefficients by g N Y ,...,Y m X ,...,X n = X Z ,...,Z g ∈ Irr( C ) N Z ,...,Z g X ,...,X n N Y ,...,Y m Z ,...,Z g . (3)Then we have the following generalized Verlinde formula . This formula is well known toexperts, but we could not find it recorded in the literature, so we provide an easy proof. Proposition 2.2.
We have g N Y ,...,Y m X ,...,X n = M C ( g, X , . . . , X n , ¯ Y , . . . , ¯ Y m ) . Proof.
In the modular functor associated to our modular category, N ZX,Y is the dimensionof the vector space assigned to a genus zero surface with two incoming and one outgoinginsertions labelled by
X, Y, Z , respectively. By gluing along punctures we can obtain numberof insertions for any genus. From the Verlinde formula [Tur94, BK01] for modular tensorcategories N X ,X ,X = 1dim C X Y ∈ Irr( C ) S X ,Y S X ,Y S X ,Y S ,Y and “sewing”, i.e. applying X Y ∈ Irr( C ) S X,Y S ¯ Z,Y = dim
C · δ X,Z . we first get the genus zero n -point Verlinde formula N X ,...,X n = 1dim C X Y ∈ Irr( C ) S X ,Y · · · S X n ,Y S n − ,Y . Applying sewing to X Z ,...,Z g ∈ Irr( C ) N Z ,...,Z g X ,...,X n N Z ,...,Z g = 1(dim C ) X Z ,...,Z g ,U,V ∈ Irr( C ) S X ,U · · · S X n ,U S ¯ Z ,U · · · S ¯ Z g ,U S n + g − ,U S Z ,V · · · S Z g ,V S g − ,V gives (2). (cid:3) The next two subsections review the basics of categorical actions and equivariantizationwhich we will need in the sequel, and the extension theory of [ENO10]. .2. Equivariantization.
We now recall some facts about equivariantizations of fusion cat-egories. As a general reference see [EGNO15, BN13]. Let C be any fusion category, and G afinite group. We will recall some notions related to categorical actions of G on C : • G is the monoidal category whose objects are elements of G and the only morphismsare identites. The monoidal product of objects is the product in the group. • Aut ⊗ ( C ) is the monoidal category whose objects are monoidal equivalences, andwhose morphisms are monoidal natural isomorphisms. The monoidal product of ob-jects is composition of functors, and the monoidal product of natural isomorphismsis the usual one. • If C is braided, then Aut br ⊗ ( C ) is the full monoidal subcategory of Aut ⊗ ( C ) whoseobjects preserve the braiding. • A categorical action is a monoidal functor G → Aut ⊗ ( C ). • If C is braided, a braided categorical action is a monoidal functor G → Aut br ⊗ ( C ) Notation for categorical actions.
In what follows below, given a categorical action, wetypically denote the functor assigned to g simply by g ( · ). The tensorator for g is typicallyindicated by ρ gX,Y : g ( X ) ⊗ g ( Y ) → g ( X ⊗ Y ). The tensorator for the categorical action isusually written µ g,h = { µ Xg,h : g ( h ( X )) → gh ( X ) } X ∈C . Given an arbitrary categorical action, recall its equivarirantization C G is defined as follows: • Objects are pairs (
X, λ ) where λ = { λ h : h ( X ) ∼ = X } h ∈ G is a family of isomorphismssatisfying g ( h ( X )) ( gh )( X ) g ( x ) x g ( λ h ) µ Xg,h λ gh λ g • Morphisms from (
X, λ ) to (
Y, δ ) consist of f ∈ C ( x, y ) such that δ g g ( f ) = λ g f for all g ∈ G .There is a canonical monoidal structure on this category which makes C G a fusion categoryif C is.Let F G : C G → C denote the forgetful functor, which simply forgets the equivariant struc-ture. In this section we provide an explicit realization of a left adjoint functor, which we willcall the induction functor I G : C → C G . On objects, we define I G ( X ) := ( L g ∈ G g ( X ) , η X )where the equivariant structure η X = { η Xh } h ∈ G is given by η Xh = M g ∈ G µ Xh,g : h ( M g ∈ G g ( X )) = M g ∈ G h ( g ( X )) → M g ∈ G hg ( X ) = M g ∈ G g ( X ) . or a morphism, f ∈ C ( X, Y ), we simply define I G ( f ) := M g ∈ G g ( f )We wish to establish I G as a left adjoint to F G . Proposition 2.3. I G as defined above is a (left) adjoint to the forgetful functor F G : C G → C Proof.
It suffices to establish a bijection C G ( I G ( X ) , ( Y, λ )) ∼ = C ( X, Y ) natural in both X and( Y, λ ).Let f ∈ C G ( I G ( X ) , ( Y, λ )). Then as a morphism in C , we may write f = L g ∈ G f g , where f g : g ( X ) → Y . Our bijection will be defined by sending f f . Since f is equivariant,we see that f gh = λ g ◦ g ( f h ) ◦ ( µ Xg,h ) − and in particular, f g = λ g ◦ g ( f ) ◦ ( µ Xg, ) − andso f is uniquely determined by f . Furthermore, for any choice of f ∈ C ( X, Y ), defining f g = λ g ◦ g ( f ) ◦ ( µ Xg, ) − yields an equivariant and setting f = L g ∈ G f g yields an equivariantmorphism. This yields the bijection. Naturality in both variables is clear. (cid:3) Since F G is monoidal, I G is both lax and oplax monoidal. In particular, we have a canonical“tensorator” ν X,Y : I G ( X ) ⊗ I G ( Y ) → I G ( X ⊗ Y ) which we describe as follows:Define ( ν X,Y ) kg,h : g ( X ) ⊗ h ( Y ) → k ( X ⊗ Y ) by( ν X,Y ) kg,h := δ g,h δ g,k ρ gX,Y , where ρ gx,y is the tensorator for the monoidal functor g . Then set ν X,Y := M g,h,k ( ν X,Y ) kg,h Furthermore, the “unit” map of I G is given by a morphism u : C G → I G ( C ), u = ⊕ g ∈ G g (1 ) . Similarly, we have a canonical ”cotensorator” ν ′ X,Y : I G ( X ⊗ Y ) → I G ( X ) ⊗ I G ( Y ) given by ν ′ X,Y := M g,h,k ∈ G ( ν ′ X,Y ) g,hk where ( ν ′ X,Y ) g,hk := δ k,g δ k,h ( ρ gX,Y ) − . It’s easy to verify that ν and ν ′ equip I G with the structure of a “special Frobenius functor”[DP08].Now let ( A, m, ι ) be an algebra object, with multiplication m : A ⊗ A → A and unit ι : → A . The tensorator ν on I G allows us to define the algebra structure ( I G ( A ) , I G ( m ) ◦ ν A,A , I G ( ι ) ◦ u ). If A in addition comes with a coproduct m ′ : A → A ⊗ A making it into aspecial Frobenius algebra, then ν ′ A,A ◦ I G ( m ′ ) makes I G ( A ) into a special Frobenius algebra. .3. Extension theory.
In this section, we briefly review the extension theory from [ENO10].Let G be a finite group. A (faithful) G -grading of a fusion category C is a decomposition aslinear categories C = L g ∈ G C g such that C g ⊗ C h ⊆ C gh and C g = 0 for all g ∈ G . If C is afusion category, a G-graded extension of C is a G -graded fusion category D = M g ∈ G D g with D = C . Theorem 2.4. [ENO10, Theorem 7.7] (Faithful) G-graded extensions of a fixed fusion cat-egory C are classified by monoidal 2-functors G → BrPic( C ) . Here G is the monoidal 2-category whose objects are elements of G and the 1 and 2morphisms are all identites. The monoidal product is given by group multiplication onobjects, and the obvious composition of identities. BrPic( C ) is the monoidal 2-categorywhose objects are invertible bimodule categories, 1-morphisms are bimodule equivalences,and 2-morphisms are bimodule functor natural isomorphisms. The monoidal product isdefined by taking the relative product of bimodules (functors, natural transformations) over C [ENO10, Definition 4.5].This classification is fairly transparent. The table below gives a correspondences betweenthe data of a monoidal 2-functor and the data of the extension (in the table below we neglectunits). Data of monoidal 2-functor Data of extension
C ⊆ D
Assignment g
7→ D g Definition of g components D = L g D g Bimodule equivalences T g,h : D g ⊠ C D h ∼ = D gh Definition of tensor product bi-functor ⊗ : D g ⊠ D h → D gh a g,h,k : T gh,k ◦ ( T g,h ⊠ C Id k ) ∼ = T g,hk ◦ (Id g ⊠ C T h,k ) Associator α : ( X g ⊗ Y h ) ⊗ Z k → X g ⊗ ( Y h ⊗ Z k )It is then shown that the coherence that the a g,h,K is required to satisfy is equivalent tothe pentagon axiom for the corresponding associator.While this result is straightforward, for it to be useful requires an understanding of themonoidal 2-category BrPic( C ), which in general is a complicated beast. However, if wetruncate the top level and take isomorphism classes of bimodule equivalence as 1-morphisms,we obtain the monoidal category BrPic( C ).This monoidal category is easier to understand. Given an invertible bimodule category M ,we have two equivalences L M , R M : Z ( C ) ∼ = End C−C ( M ) given by left and right multiplicationrespectively. The composition L − M ◦ R M gives a braided auto-equivalence α M ∈ Aut br ⊗ ( Z ( C )). Theorem 2.5. [ENO10, Theorem 1.1]
The assignment
M 7→ α M described above extendsto a monoidal equivalence BrPic( C ) ∼ = Aut br ⊗ ( Z ( C )) . Thus given an extension with classifying functor ρ : G → BrPic( C ), decategorifying canon-ically gives a monoidal functor ρ : G → BrPic( C ) ∼ = Aut br ⊗ ( Z ( C )) . he goal of this paper, is to recover the fusion rules of the extension from just the cat-egorical action ρ . The reason this is useful is that often we use the above theorems in thereverse direction.Namely, suppose we want to construct and extension from scratch. Then we can start froma categorical action ρ : G → Aut br ⊗ ( Z ( C )) ∼ = BrPic( C ). Then we can lift this to a monoidal2-functor ρ : G → BrPic( C ) if and only if a certain obstruction o ( ρ ) ∈ H ( G, C × ) vanishes.If it does, then we know an extension exists, and the possible associators form a torsor over H ( G, C × ).In practice, using this method one can often show the o obstruction vanishes for generalreasons (for example [GJ19]). In this situation, we know extensions exists, but it is often verydifficult to say anything about the structure of such extensions in general. Thus new methodsare required to work out the details of what an extension looks like when constructed in thisway. The goal of this paper is precisely to provide such methods to determine the fusionrules of the extension.In the sequel our method will require the existence of a spherical structure on the extension D . As mention in the introduction, to our knowledge there has been no general theorydeveloped for constructing spherical structures on extensions though it should certainly exist.For example, one may naively guess that if the categorical action ρ is spherical, then all theresulting extensions will have a canonical spherical structure. Thus having conditions on C which would guarantee the existence of a spherical structure on our extensions automaticallywould make it easier to apply apply our results.We have the following proposition. Proposition 2.6.
Let C be a pseudo-unitary fusion category, and let C ⊆ D be a G -gradedextension. Then D is pseudo-unitary.Proof. Let X ∈ D be in the g -graded component. Then choose a dual object ¯ X ∈ D g − ,and solutions to the duality equations R X , ¯ R X , R ∗ X , ¯ R ∗ X (here we use the notation precedingequation 1). Then X ⊗ ¯ X ∈ C is canonically equipped with the structure of a connectedspecial Frobenius algebra, with multiplication m := 1 X ⊗ R ∗ X ⊗ ¯ X , co-multiplication ∆ :=1 X ⊗ R X ⊗ ¯ X , unit ι := ¯ R X and counit ǫ := ¯ R ∗ X .Then this algebra is special, with constants m ◦ ∆ = ( R ∗ X · R X )1 X ⊗ ¯ X and ǫ ◦ ι = ¯ R ∗ X · ¯ R X .Thus the invariant quantity β associated to any special Frobenius algebra defined by ǫ · m · ∆ · ι = β in this case is precisely the paired dimension d { X, ¯ X } .Since X is simple, the algebra X ⊗ ¯ X is connected (also called haploid in the literature).Thus by [FRS02, Corollary 3.10], this algebra will be symmetric with respect to any sphericalstructure for which the spherical dimension of X ⊗ ¯ X is non-zero. But symmetric specialFrobenius algebras A satisfy β = d A . In particular, choosing the canonical pseudo-unitaryspherical structure, the above shows connected Frobenius algebra X ⊗ ¯ X is symmetric, hence d { X, ¯ X } = β = d X ⊗ ¯ X = FPdim C ( X ⊗ ¯ X ) = FPdim D ( X ⊗ ¯ X ) = FPdim D ( X ) . Thus D ispseudo-unitary. (cid:3) Recovering fusion rules from the Lagrangian algebra
In this section, we will explain how the fusion rules of a fusion category can be derivedfrom a pair of algebraic operations on the vector space End Z ( C ) ( I ( )). We use these resultstogether with the facts we’ve assembled about equivariantizations to describe the fusion rules or extensions. Our conventions for half-braidings and spherical structures follow [M¨ug03a],[M¨ug03b]. We use the graphical calculus for spherical fusion categories freely.We refer the reader to [FRS02] for definitions concerning algebras in tensor categoriesand their various adjectives. We warn the reader that following [EGNO15] we use the word connected to mean dim( C ( , A )) = 1, whereas in many references (including [FRS02]) theword haploid is used. Let A be any commutative, connected special Frobenius algebra in abraided spherical fusion category with d A = 0, normalized so that AA = AA , ε · i = A = d A . Then we can define the convolution product on End C ( A ) by AAa ∗ b := AAa b . (4)This operation on End C ( A ) makes it into an associative, commutative algebra. The unit with respect to the convolution product is given by i ◦ ε = AA We note that End C ( A ) also has the usual composition, and thus we have two operationson this vector space (End C ( A ) , ◦ , ∗ ). By [BD18, Corollary 2.5], (End C ( A ) , ∗ ) is a semi-simple commutative algebra and is thus isomorphic to C n . Thus we can “diagonalize” themultiplication by finding minimial idempotents. We note this idempotents give a canonicalbasis for the vector space End C ( A ). Lemma 3.1.
Let
A, B be connected special Frobenius algebras with non-zero dimension, nor-malized as above, which are isomorphic as algebras. Then (End C ( A ) , ◦ , ∗ ) ∼ = (End C ( B ) , ◦ , ∗ ) .Proof. By [FRS02, Corollary 3.10], A and B are symmetric, hence by [FRS02, Theorem3.6] there is a unique comultiplication with the desired normalization. Therefore any alge-bra intertwiner ψ ∈ C ( A, B ) must also intertwine the comultiplications. Indeed, if m B ∈C ( B ⊗ B, B ) and n B ∈ C ( B, B ⊗ B ) denote the normalized Frobenius multiplication andcomultiplication for B respectively, then ( ψ − ⊗ ψ − ) ◦ n B ◦ ψ ∈ C ( A, A ⊗ A ) provides anappropriately normalized comultiplication for m A and therefore must be n A (a similar argu-ment applies to counits). Thus the map End C ( A ) → End C ( B ), f ψ ◦ f ◦ ψ − ∈ End C ( B )is an isomorphism with respect to ◦ and ∗ . (cid:3) .1. The canonical Lagrangian algebra.
Recall that on object in the Drinfeld center Z ( C ) consists of pairs ( Y, φ Y ) where Y ∈ C and φ Y is a natural isomorphisms from thefunctor Y ⊗ · → · ⊗ Y called half-braidings satisfying a version of the hexagon coherence[M¨ug03b]. Morphisms between such pairs consist of morphisms between the underlyingobjects which intertwine the half-braidings. The functor from Z ( C ) to C which sends a pair( Y, φ Y ) to the object Y and morphisms to themselves is called the forgetful functor , denotedF : Z ( C ) → C .Let C be a spherical fusion. Let us once and for all pick a square root √ d X for each X ∈ Irr( C ). The forgetful functor admits a (left) adjoint I : C → Z ( C ). By [KJB10], we canrepresent I with the following explicit formula: I ( X ) := ( L Y ∈ Irr( C ) Y ⊗ X ⊗ Y ∗ , φ I ( X ) ) ,φ I ( X ) ,W := M X,Y ∈ Irr( C ) X i p d Y p d Z YZ XXW ¯ Y ¯ Z Wi i • , Here { i } is a basis for C ( Y, W ⊗ Z ) and { i • } ⊆ C ( ¯ Y ⊗ W, ¯ Z ) is a dual basis with respect tothe pairing h j, i i := ¯ Y W Zij . (5)In [KJB10, Theorem 2.3], the authors establish I as a (left) adjoint to the forgetful func-tor. In particular, they provide a canonical bijection C ( X, F ( Y, φ Y )) ∼ = Z ( C )( I ( X ) , ( Y, φ Y ))defined by XYf M Z ∈ Irr( C ) p d Z XYf φ Y, ¯ Z Z ¯ Z . (6)The object I ( ) is canonically endowed with the structure of a (symmetric) special Frobeniusalgebra in Z ( C ), with structure maps A AA = M X ∈ Irr( C ) √ d X ¯ X XX X ¯ X ¯ X , A = M X ∈ Irr( C ) p d X ¯ XX The comultiplication and counit are given by the reflected diagrams of the multiplicationand unit maps respectively, with the same normalizing coefficients. Thus I ( ) is a connected pecial Frobenius algebra normalized as in the previous section (note d I ( ) = P X ∈ Irr( C ) d X =dim( C ) > Z ( C )( , F ◦ I ( )) ∼ = M X ∈ X C ( , X ⊗ ¯ X ) . We have a basis for L X ∈ X C ( , X ⊗ ¯ X ) consisting of cups. Namely, set r Y := p d Y Y ¯ Y . Then { r Y } form a basis for L X ∈ X C ( , X ⊗ ¯ X ). Now we consider the image of r Y under thecanonical adjunction from equation 6, given by e Y = M X,Z ∈ Irr( C ) X i d Y p d X p d Z XZ Y ¯ X ¯ Zi • i (7)where the summation of i is over a basis for C ( Y ⊗ ¯ X , ¯ Z ), and i • is a dual basis with respectto the obvious graphical pairing (c.f. 5, and note this pairing is rotationally invariant). Astraightforward computation then gives us the following:(1) { e Y } forms a basis for End Z ( C ) ( I ( )),(2) e Y ∗ e Z = δ Y,Z e Y .In other words, the collection { e Y } diagonalizes the convolution product. Proposition 3.2. e Y ◦ e Z = P X ∈ Irr( C ) d Y d Z d X N XY Z e X .Proof. We see that e Y ◦ e Z = M P,R ∈ Irr( C ) X Q ∈ Irr( C ) X i,j d Y d Z d Q p d R p d P PR ZY ¯ P ¯ Ri • ij • jQ ¯ Q . However, the sets d Q Y ¯ RZ ¯ P ¯ Qj i , d
U Y ¯ RZ ¯ PU kl as Q, U runs over Irr( C ) and i, j, k, l run over the graphically normalized bases both form a(graphically normalized) basis for C ( Y ⊗ Z ⊗ ¯ P , ¯ R ). Since the first basis set appears together ith its dual in the above expression, we can replace it with the latter, to obtain e Y ◦ e Z = M P,R ∈ Irr( C ) X i d Y d Z N UY,Z p d R p d P PR U ¯ P ¯ Rk • k = X U ∈ Irr( C ) d Y d Z d U N uY,Z e U . (cid:3) Now let C be a fusion category and d a spherical dimension function. Then consider( K ( C ) , · , ∗ d ), where K ( C ) = C [Irr( C )], [ X ] · [ Y ] = P Z ∈ Irr( C ) N ZXY [ Z ], and [ X ] ∗ d [ Y ] = δ X,Y d X [ X ]. Then the above proposition and a straightforward computation gives us the follow-ing corollary: Corollary 3.3.
The assignment e X d X [ X ] gives an isomorphism (End Z ( C ) ( I ( )) , ◦ , ∗ ) ∼ = ( K ( C ) , · , ∗ d ) . Thus if we have the algebraic structure (End Z ( C ) ( I ( )) , ◦ , ∗ ) and we know the sphericaldimension function , we can determine the fusion rules by rescaling the canonical basis.Unfortunately this is not information we will have a-priori.In the extension construction described in Section 2.3 the input is a categorical action ρ : G → Aut br ⊗ ( Z ( C )). Suppose o ( ρ ) vanishes, so there exists a (several) extension C ⊆D . We would like to compute the fusion rules for this extension. We will assume theextension D admits a spherical structure. In the next section we will show how to compute(End Z ( D ) ( I ( )) , ◦ , ∗ ).As we’ve mentioned, a-priori this is not quite enough to reconstruct the fusion rules, sincewe don’t know which dimension function d our basis is scaled with respect to! Indeed, wedo not even know the fusion rules of D yet, so trying to determine the possible dimensionfunctions is premature.However, we can determine the square of the dimensions (i.e. the paired dimensions) asfollows: first determine the canonical basis element acting as the unit under composition, e which is straightforward. For each e Y , there will be a unique element e ¯ Y such that e Y e ¯ Y hasa coefficient of e . This coefficient will be d Y = d { Y, ¯ Y } >
0, the canonical paired categoricaldimension (see equation 1). Recall the positive square root (i.e. the fusion dimension)is denoted d + Y . We have d Y = γ Y d + Y , where γ · : Irr( D ) → {± } is determined by (anddetermines) the spherical structure. More explicitly, to have a spherical structure we needto find a function γ · as above such that whenever Z ≺ X ⊗ Y , we have γ X γ Y γ Z = T ZXY is thepivotal operator (See [Bar16, Theorem 5.4]). In particular, for such a spherical structuredefined by γ · to exist, we must have T ZXY = ± Z ≺ X ⊗ Y .In any case, when we have a spherical structure, the spherical dimensions necessarilysatisfy d X d Y d Z = ± d + X d + Y d + Z . Having the abstract algebra and the scaled basis elements, we can compute the coefficientof e Z in e X e Y , which is C ZXY = d X d Y d Z N ZXY . We can also determine d + X d + Y d + Z as described above, nd thus the fusion rule can be recovered as N ZXY = (cid:12)(cid:12)(cid:12)(cid:12) C ZX,Y d + Z d + X d + Y (cid:12)(cid:12)(cid:12)(cid:12) . (8) Summary of preceding discussion.
Suppose we are given ( V, ◦ , ∗ ) which we know is iso-morphic to ( K ( C ) , · , ∗ d ) for some spherical dimension function d . Using the above procedurewe can recover the fusion rules without determining d . While our method requires the exis-tence of a spherical structure to produce the fusion rules, it does not require the choice of aspecific one Remark 3.4.
Above we assumed the existence of a spherical structure to derive our result. Inthe hypothetical case that there exists a fusion category which admits no spherical structure,the convolution product and composition product still make sense for the Frobenius algebra I ( ) . On can show, however, that instead of recovering the fusion rules using our procedureabove, we recover the signed fusion rules T r ( T ZXY ) , where again T ZXY is the pivotal operator [Bar16] . It does not seem to be possible to recover the fusion rules from this informationunless T ZXY = ± for every triple of simple objects. G -extensions. In the previous section, we showed how to recover the fusion rules of afusion category from the algebraic structure of the Lagrangian algebra I ( ) ∈ Z ( C ). Givena G -extension C ⊆ D , and have a categorical action ρ : G → Aut br ⊗ ( Z ( C )). The point ofthis section is to show how to describe the endomorphisms, convolution, and compositionproduct of the canonical Lagrangian algebra for D in terms of the data of the Lagrangianalgebra for C and the categorical action ρ . Our approach is based on the results of [GNN09],which realize the Drinfeld center Z ( D ) as a certain equivariantization.Recall from Section 2.3 that given a G -extension C ⊆ D , we have a canonically associatedcategorical action ρ : G → Aut br ⊗ ( Z ( C )). From [GNN09, Theorem 3.3], we have that therelative center of the extension Z C ( D ) is a G -crossed braided extension of Z ( C ), whose G -action on the trivial component is precisely the canonical action ρ .Furthermore, we have Z ( D ) = Z C ( D ) G [GNN09, Theorem 3.5]. The forgetful functor Z ( D ) → D factorizes as the composition of the forgetful functor Z ( D ) → Z C ( D ) with Z C ( D ) → D , and thus its (right) adjoint factors as a composite of the respective adjoints.However, upon identification of Z ( D ) with Z C ( D ) G , the first forgetful functor (which forgetsthe half-braiding with all D and just remembers the half-braiding with the trivial component)is identified with the functor that forgets the equivariant structures on objects F G which wedescribed above.Denote I D : D → Z ( D ) the (left) adjoint of the forgetful functor, I C : C → Z ( C ) the (right)adjoint of the forgetful functor and I G : Z C ( D ) → Z C ( D ) G ∼ = Z ( C G ) as above. Then we have I D ∼ = I G ◦ I C . In particular, I D ( ) = I G ( I C ( )). Thus the description of the algebra structure on I G ( A )(for an arbitrary algebra A ) from Section 2.2 provides a model for the canonical Lagrangianalgebra I D ( ). o compute the fusion rules, our first step is to identify End( I D ( )) as a vector space. Wesee Z ( D )( I D ( ) , I D ( )) ∼ = Z ( C ) G ( I G ( I C ( )) , I G ( I C ( ))) ∼ = M g ∈ G Z ( C )( I C ( ) , g ( I C ( )))where the last isomorphisms uses the model for I G and the adjunction from Proposition2.3. Let L := I C ( ), with multiplication m and comultiplication m ′ as described above.Then using the description of the adjunction to transport the convolution and compositionstructures from I G ( I C ( )), we have the following description: K ( D ) ∼ = M g ∈ G Z ( C )( L, g − ( L )) . For a g ∈ Z ( C )( L, g − ( L ) , b h ∈ Z ( C )( L, h − ( L )) a g ∗ b h := δ g,h g − ( m ) ◦ ρ g − L,L ◦ ( a g ⊗ b g ) ◦ m ′ ∈ Z ( C )( L, g − ( L )) . (9)For the composition product, we see a g ◦ b h := µ Lh − ,g − ◦ h − ( a g ) ◦ b h ∈ Z ( C )( L, ( gh ) − ( L )) . (10) Remark 3.5.
Note that while we use ◦ for the composition product above, this is an abuseof notation, and is not actually the operation of composition of the morphisms a g and b h in the category Z ( C ) . Indeed this doesn’t even make sense in general since they have dif-ferent sources and targets. Rather, this operation corresponds to honest composition of theendomorphisms of I G ( L ) obtained by applying the adjunction from Proposition 2.3. We now put everything together to describe an algorithm for finding the fusion coefficientsof a G -extension of a fusion category: Algorithm for finding fusion rules of G -extension: (1) First find arbitrary basis B g for V g := Z ( C )( L, g − ( L )) for each g ∈ G .(2) Compute convolution product ∗ (see equation 9) and composition product ◦ (seeequation 10) in terms of the basis S g ∈ G B g .(3) Find minimal projections of V g with respect to convolution, label them e Y . Thesewill correspond to simple object in the g component of the extension.(4) Next we want to compute the C ZXY in the sum e X ◦ e Y = P Z C ZXY e Z . To do this, weuse ( e X ◦ e Y ) ∗ e Z = C ZXY e Z .(5) Next, we note that for each e Y ∈ V g , there is a unique e ¯ Y ∈ V g − such that C Y ¯ Y > d + X = q C Y ¯ Y .(6) We then determine N ZXY = (cid:12)(cid:12)(cid:12)(cid:12) C ZXY d + Z d + X d + Y (cid:12)(cid:12)(cid:12)(cid:12) . .3. Example: Fusion categories Morita equivalent to
Vec( A ) . Let A be an abeliangroup. Then Z (Vec( A )) ∼ = Vec( ˆ A × A, q ), where A is an abelian group and q ( ϕ, a ) = ϕ ( a )the canonical quadratic form on ˆ A × A , where ˆ A = Hom( A, C × ) is the dual group [ENO10].Fusion categories C together with a Morita equivalence to Vec( A ) are described by La-grangian algebras in Z (Vec( A )). The Lagrangian algebra is precisely I C ( ).Lagrangian algebras in Vec( ˆ A × A, q ) correspond precisely to Lagrangian subgroups L ≤ ( ˆ A × A, q ) [DS18]. By definition, these are precisely the subgroups with | L | = | A | such that q | L = 1. By [ENO10, Proposition 10.3], these are in bijective correspondence subgroups H ≤ A together with alternating bicharacters (which are, alternatively, in bijective correspondencewith elements of H ( H, C × )).Given H ≤ A and b ∈ Alt( H × H, C × ) define L h,b = { ( ϕ, h ) ∈ ˆ A × A : ϕ ↾ H = b ( h, · ) } and consider the Lagrangian subgroup L H,b = [ h ∈ H L h,b . then it follows that ( H, b ) L H,b is a one-to-one correspondence between pairs (
H, b ) asabove and Lagrangian subgroups L ≤ ( ˆ A × A, q ). Namely, given L ≤ ( ˆ A × A, q ) Lagrangian,define H by { } × H = L ∩ ( { } × A ) and b ( h, k ) = ϕ h ( k ) for some ( ϕ h , h ) ∈ L .Now, for arbitrary a ∈ ˆ A × A , let χ a be the character on ˆ A × A defined by χ a ( b ) := q ( ba ) q ( b ) q ( a ) . Suppose we have a homorphism π : G → O ( ˆ A × A, q ), i.e. a homomorphism π : G → Aut( ˆ A × A ) such that q ( g ( a )) = q ( a ), where by abuse of notation we denote g ( · ) = π ( g )( · ). Let ω : G × G → b A × A be a 2-cocycle with respect to this homomorphism, i.e. ω g,hk g ( ω h,k ) = ω g,h ω gh,k . Then this data defines a braided categorical action π ω : G → Aut br ⊗ ( C ( ˆ A × A, q )) , where the element g acts by π ( g ) in the obvious way as a strict monoidal functor on Vec( b A × A ). We again abuse notation and use g ( · ) to refer to the functor π ( g ). To define thetensorator of the categorical action we use the monoidal natural isomorphisms µ ag,h := χ ω g,h ( gh ( a ))1 gh ( a ) : g ( h ( a )) = gh ( a ) → gh ( a ) . That this is a categorical action follows from the general theory of [ENO10]. However, forthe sake of completeness we give a direct verification.We need to verify for all a ∈ ˆ A × A , g, h, k ∈ Gµ agh,k µ k ( a ) g,h = µ ag,hk g ( µ ah,k )Using our definition of µ ag,h , this becomes χ ω gh,k ( ghk ( a )) χ ω g,h ( ghk ( a )) = χ ω g,hk ( ghk ( a )) χ ω h,k ( hk ( a )) . (11) ut since g preserves q we have χ a ( g ( b )) = χ g − ( a ) ( b ). Thus we take the left hand side ofequation 11, and we compute χ ω gh,k ( ghk ( a )) χ ω g,h ( ghk ( a )) = χ ghk − ( ω gh,k ) ( a ) χ ghk − ( ω g,h ) ( a )= χ ghk − ( ω gh,k ω g,h ) ( a )= χ ghk − ( ω g,hk g ( ω h,k )) ( a )= χ ω g,hk ( ghk ( a )) χ ω h,k ( hk ( a ))as desired. It turns out every braided categorical action on Vec( ˆ A × A, q ) is equivalent toone of this form [ENO10].Let L ≤ ˆ A × A be a Lagrangian subgroup, and by an abuse of notation, let L also denotethe corresponding Lagrangian algebra.Then as an object, L = L a ∈ L a . The multiplication is given by m := 1 p | A | M a,b m a,b : M a,b ∈ L a ⊗ b → M c ∈ L c, where m a,b = 1 a ⊗ b . The Frobenius comultiplication is defined similarly.
Theorem 3.6.
Let π : G → Aut( ˆ A × A, q ) be a group homomorphism and ω an ˆ A × A -valued2-cocycle, and consider the categorical action constructed from this data as described above.Let L be a Lagrangian subgroup and set L g := L ∩ g − ( L ) . Then the simple objects in the G -graded component of any corresponding extension of Vec( ˆ A × A, q ) L (if it exists) are indexedby irreducible characters α ∈ c L g . For α ∈ c L g , β ∈ c L h , γ ∈ d L gh we have N γαβ = δ αβχ ωh − ,g − | Lg ∩ Lh ,γ | Lg ∩ Lh | L g ∩ L h | p | A | p | L g | | L h | | L gh | . Proof.
Note that C ( ˆ A × A, q ) L ∼ = Vec( ˆ L, µ ) for some 3-cocycle µ ∈ Z ( ˆ L, C × ). All thesecategories are pseudo-unitary, hence we can apply our algorithm to any extension.First we compute the convolution structure. A basis for Vec( ˆ A × A )( L, g − ( L )) is givenby { a } a ∈ L ∩ g ( L ) , a ∗ b = 1 | A | ab . Let α, β ∈ c L g be irreducible characters. Then we have the standard formula from charactertheory X a ∈ L g α ( a ) β ( a − b ) = δ α,β | L g | α ( b ) . Thus we may define e α = | A || L g | M a ∈ L g α ( a )1 a and hence e α ∗ e β = δ α,β e α . or α ∈ c L g , β ∈ c L h , γ ∈ d L gh we compute( e α ◦ e β ) ∗ e γ = | A | | L g || L h || L gh | M b ∈ L gh X a ∈ L g ∩ L h χ ω h − ,g − ( a ) α ( a ) β ( a ) γ ( a − b ) b = C γαβ e γ . We know the coefficient of 1 in e γ is precisely | A || L gh | .Using the character formula X a ∈ L g ∩ L h χ ω h − ,g − ( a ) α ( a ) β ( a ) γ ( a − b ) = | L g ∩ L h | δ αβχ ωh − ,g − | L g ∩ L h , γ | L g ∩ L h γ ( b )and comparing coefficients of 1 via our above expression, we obtain C γαβ = | L g ∩ L h || A || L g || L h | δ αβχ ωh − ,g − | Lg ∩ Lh ,γ | Lg ∩ Lh . To find the multiplicities, we see the positive dimensions satisfy d + α = s | A || L g | and therefore N γαβ = δ αβχ ωh − ,g − | Lg ∩ Lh ,γ | Lg ∩ Lh | L g ∩ L h | p | A | p | L g || L h || L gh | . (cid:3) We remark that these formulas can be applied to derive the fusion rules for reflectionfusion categories [EG18] in the case when the trivial component of the category (which isan elementary abelian p-group) has trivial 3-cocycle associator.3.4.
Example: Fusion categories with center tensor equivalent to
Vec( B ) . We canslighlty generalize the former example. Let A be an abelian group. We want to consider thefollowing kind of Lagrangian extensions of A . Let B be another abelian group with | B | = | A | and b : B × B → C × a bicharacter, such that q : B → C × defined by q ( x ) = b ( x, x ) is anon-degenerate quadratic form and that there is an embedding ˆ A ֒ → B with q | ˆ A ≡
1. Notethat [LN14, Lemma 4.4] implies that the modular tensor category C ( B, q ) is monoidallyequivalent to Vec( B ). The Lagrangian subgroup ˆ A ≤ B gives a Lagrangian algebra L in C ( B, q ). We have that C ( B, q ) L is tensor equivalent to Vec( A, µ ) for some µ ∈ H ( A, C × )and C ( B, q ) is braided equivalent to Z (Vec( A, µ )). For a ∈ B let χ a to be the character on B defined by χ a ( g ) := q ( ag ) q ( a ) q ( g ) = b ( a, g ) b ( g, a ) . Suppose as above, we have a homorphism π : G → O ( B, q ) and ω : G × G → B a 2-coyclewith respect to this homorphism as above. By replacing ˆ A × A by B we get a categoricalaction of G on C ( B, q ) as before. All the arguments are the same replacing ˆ A × A by B , thuswe get the slightly more general version of Theorem 3.6: Theorem 3.7.
With the above notation, set L g := L ∩ g − ( L ) and A g = A/ { a ∈ A | ev a | L g =id L g } . Then the simple objects in the G -graded component of the corresponding extension f Vec(
A, µ ) ∼ = C ( B, q ) L (if it exists) are indexed by irreducible characters α ∈ A g . For α ∈ A g , β ∈ A h , γ ∈ A gh we have N γαβ = δ αβχ ωh − ,g − | Lg ∩ Lh ,γ | Ag ∩ Ah | A g ∩ A h | p | A | p | A g | | A h | | A gh | . We remark that theorem applies to any C whose center Z ( C ) is tensor equivalent to Vec( B ).Namely, in this case Z ( C ) is braided equivalent to C ( B, q ) where q is a quadratic form on B which comes from a bicharacter b on B by [LN14, Lemma 4.4]. Then I ( ) ∈ C ( B, q )gives a Lagrangian subgroup L and C ( B, q ) L is tensor equivalent to Vec( A, µ ) for some µ ∈ H ( A, C × ), where A = ˆ L . In particular, it applies to C = Vec( A, µ ) where A is of oddorder and µ is a “soft” cocycle. Here the subgroup of “soft” cohomology classes is H ( A, C × ) ab = ( [ ω ] ∈ H ( A, C × ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y σ ∈ S ω ( σ ( x ) , σ ( y ) , σ ( z )) sign σ = 1 for all x, y, z ∈ A ) . By [MN01, Corollary 3.6], Z (Vec( A, µ )) is pointed for every [ µ ] ∈ H ( A, C × ) ab thus braidedequivalent to some C ( B, q ). If A is odd, define a bicharacter b on B by b ( g, h ) = (cid:18) q ( gh ) q ( g ) q ( h ) (cid:19) Exp( G )+12 then q ( g ) = b ( g, g ). Thus for every A odd abelian group and [ µ ] ∈ H ( A, C × ) ab the categoryVec( A, µ ) arise in the above way.4.
Examples from G -crossed extensions of modular categories We turn our attention to the case C is modular. Then Z ( C ) is braided equivalent to C ⊠ C rev , and the forgetful functor is the functor X ⊠ Y X ⊗ Y ∈ C . We first give andescription of L = I ( ) in C ⊠ C rev . By [KR08, Section 2.2], we can describe the canonicalLagrangian algebra as follows:As an object L = M X ∈ Irr( C ) X ⊠ ¯ X .
The multiplication is given by m = M X,Y,Z ∈ Irr( C ) X i Z YiX ⊠ ¯ Z ¯ Y ˇ i ¯ X , here { i } ⊆ C ( X ⊗ Y, Z ) consists of a basis for C ( X ⊗ Y, Z ), and { ˇ i } ⊆ C ( ¯ X ⊗ ¯ Y , ¯ Z ) is abasis given by ¯ Z ¯ Y ˇ i ¯ X = i ∗ ¯ X ¯ Z ¯ Y . where i ∗ is the dual basis with respect to the composition pairing. The unit is given by ι = 1 ⊠ . Now, by Lemma 3.1, for the purposes of computing the hypergroup, we can replace thisalgebra by any isomorphic algebra which also admits a symmetric Frobenius algebra with ourdesired normalization. In particular, we can choose one for which the normalized Frobeniuscomultiplication is easier to compute.Thus we consider the same object L , but with multiplication m := 1 p dim( C ) M X,Y,Z ∈ Irr( C ) √ d X d Y √ d Z X i i ⊠ ˇ i with ˇ i defined as above, and unit ι = p dim( C )1 ⊠ . Then the map ψ := M Z ∈ Irr( C ) s d Z dim( C ) 1 Z ⊠ ¯ Z ∈ C ⊠ C rev ( L, L )is an automorphism of the object A , but satisfies ψ ◦ m = m ◦ ( ψ ⊗ ψ ) and ψ ◦ ι = ι , andthus ( A, m, ι ) ∼ = ( A, m , ι ).Furthermore, we can more easily compute the correctly normalized Frobenius comultipli-cation to be given by n := 1 p dim( C ) M X,Y,Z ∈ Irr( C ) r d X d Y d Z X i i ∗ ⊠ ˇ i ∗ , where { i ∗ } ⊆ C ( Z, X ⊗ Y ) is dual to { i } with respect to the composition pairing, and { ˇ i ∗ } is dual to { ˇ i } with respect to the composition pairing. Defining the counit ǫ := p dim( C )1 ⊠ we obtain a symmetric Frobenius algebra structure on ( A, m, ι ) with the correct normaliza-tion as desired.We note in the modular case, the story is considerably simplified for two reasons. First,we have (End( A ) , ◦ ) ∼ = Fun(Irr( C )) as algebras, where the latter is the algebra of complexvalued functions on the set Irr( C ) with point wise multiplication. The identification is via f ∈ Fun(Irr( C )) M X ∈ Irr( C ) f ( x )1 X ⊠ ¯ X . ith this notation, an easy computation gives( f ∗ g )( Z ) = 1dim( C ) X X,Y ∈ Irr( C ) f ( X ) g ( Y ) d X d Y d Z N ZXY . (12)In terms of the basis { X ⊠ ¯ X : X ∈ Irr( C ) } we have1 X ⊠ ¯ X ∗ Y ⊠ ¯ Y = 1dim( C ) X Z ∈ Irr( C ) N ZXY d X d Y d Z Z ⊠ ¯ Z . We will use both expressions in the sequel based on convenience.
Lemma 4.1.
Let C be a modular tensor category, then for all V ∈ Irr( C ) , then C X X,Y ∈ Irr( C ) N ZX,Y ( d V S X,V )( d W S Y,W ) = δ V,W d V S Z,V . Proof.
We have 1 d W S X,W S Y,W = X Z ∈ Irr( C ) N ZX,Y S Z,W and therefore X X,Y ∈ Irr( C ) N ZX,Y ( d V S X,V )( d W S Y,W ) = d V d W X X ∈ Irr( C ) S X,V X Y ∈ Irr( C ) N Y ¯ X,Z S Y,W = d V X X ∈ Irr( C ) S X,V S ¯ X,W S Z,W = δ V,W dim C d V S Z,V . In the last equality, we used the property of modular data
SCS = CS = dim C · I , where C = (dim C ) − · S is the charge conjugation matrix given by C X,Y = δ X, ¯ Y and I is theidentity matrix (see [BK01, Theorem 3.1.7]). (cid:3) Proposition 4.2.
The set { e V } V ∈ Irr( C ) with e V = M X ∈ Irr( C ) d V d X S X,V · X ⊠ ¯ X forms a complete set of minimal idempotents for (End Z ( C ) ( L ) , ∗ ) .Proof. Note { e V } V ∈ Irr( C ) forms a basis of End Z ( C ) ( L ) since S is invertible.Furthermore, we have e V ∗ e W = 1dim( C ) X X,Y,Z d V d X S X,V d W d Y S Y,W d X d Y d Z N ZXY Z ⊠ ¯ Z = 1dim( C ) X X,Y,Z N ZXY ( d V S X,V )( d W S Y,W ) 1 d Z Z ⊠ ¯ Z = δ V,W X Z ∈ Irr( C ) d V d Z S Z,V Z ⊠ ¯ Z = δ V,W e V . (cid:3) s a consistency check, we compute the composition structure e X ◦ e Y = M V ∈ Irr( C ) d X d Y d V S X,V S Y,V · V ⊠ ¯ V = M V ∈ Irr( C ) X Z ∈ Irr( C ) N ZX,Y d X d Y d Z d Z d V S Z,V · V ⊠ ¯ V = X Z ∈ Irr( C ) N ZX,Y d X d Y d Z · e Z . This analysis was purely of the canonical Lagrangian algebra, with no categorical action.We will now study a particular type of categorical action, which is associated to a G -crossedbraided extension of C rather than an ordinary extension.Recall for non-degenerate fusion categories that there is a monoidal functor π : Aut br ⊗ ( C ) → Aut br ⊗ ( Z ( C )) = Aut br ⊗ ( C ⊠ C rev ), where the braided C autoequivalence acts on the second factor1 ⊠ C rev ⊆ C ⊠ C rev . Furthermore, there is an equivalence ∂ : Aut br ⊗ ( C ) ∼ = Pic( C ) defined via α induction [DN13]. Then by [DN13], the following diagram commutes up to monoidal naturalisomorphism Aut br ⊗ ( C ) Aut br ⊗ ( Z ( C ))Pic( C ) BrPic( C ) ∂ π Forget where the arrow on the right is the canonical equivalence from Section 2.3. G -extensions of modular categories together with a whose canonically associated categor-ical action has a lift Aut br ⊗ ( C ) ∼ = Pic( C ) G Aut br ⊗ ( Z ( C )) ∼ = BrPic( C )correspond to G-crossed braidings on the G -extension.We will continue to consider this case in generality.Let g ∈ Aut br ⊗ ( C ), and define Fix g = { X ∈ Irr( C ) : g ( X ) ∼ = X } . Then we have a clearisomorphism Fun(Fix g ) ∼ = Z ( C )( L, g ( L )) f M X ∈ Fix g f ( X )1 X ⊠ ¯ X . Our first step is to determine a formula for the convolution product on each componentFun(Fix g ) ∼ = Z ( C )( L, g ( L )).Suppose X, Y, Z ∈ Fix g . Then choose isomorphisms γ X : g ( X ) → X , γ Y : g ( Y ) → Y , γ Z : g ( Z ) → Z , we can define a linear operator U g,ZX,Y : C ( X ⊗ Y, Z ) → C ( X ⊗ Y, Z ) α γ Z · g ( α ) · ρ gX,Y · ( γ − X ⊗ γ − Y ) . hen define T g,ZX,Y := Tr( U g,ZX,Y ) . Note that while U g,ZX,Y itself depends on the choice of γ X , γ Y , γ Z , the quantity T g,ZX,Y does not.Furthermore, from the formula for convolution (9), we obtain the following proposition: Proposition 4.3.
Let
X, Y ∈ Fix g . Then X ⊠ ¯ X ∗ Y ⊠ ¯ Y = 1dim( C ) X Z ∈ Fix g d X d Y d Z T g,ZX,Y Z ⊠ ¯ Z . Unlike the previous convolution product, this one does not seem possible to analyze atthis level of generality.4.1.
Cyclic permutation actions.
Let C be a modular tensor category and G ֒ → S n a G -space. Then there is an action of G on C ⊠ n by permutations. We denote a G -crossed braidedextension of C ⊠ n by C≀ G as in [Tur10] noting that it is not necessarily unique, but always existsby [GJ19] (see Remark 4.7). Let us consider the cyclic subgroup Z /n Z ֒ → h (12 · · · n ) i ≤ S n ,1 (12 · · · n ). As an application of our algorithm we compute the fusion rules for C ≀ Z /n Z categories.Suppose g ∈ Z /n Z . Let o denote the order of g . We define the co-order of g by c ( g ) := no ( g ) Then the simple objects in C ⊠ n fixed by g up to isomorphism are of the form X = ( X ⊠ X ⊠ · · · ⊠ X c ( g ) ) ⊠ o ( g ) where X , · · · , X c ( g ) ∈ Irr( C ) are arbitrary.Then we have the following claim: Lemma 4.4.
The set of minimal convolution idempotents is given by { f g,X } X ∈ Irr( C ⊠ m ) , where f g,X = X ⊠ ··· ⊠ X m = M Y = Y ⊠ ··· ⊠ Y m d X (dim C ) n − c ( g ) d o ( g ) Y c Y i =1 ( g ) S X i ,Y i | {z } = S X,Y · Y ⊠ o ( g ) ⊠ ¯ Y ⊠ o ( g ) . Proof.
We may assume g is a generator. Otherwise replace C by C ⊠ c ( g ) and replacing n by o ( g ). Then we have f g,X ∗ f g,Y = X U,V,W d X d Y d nU d nV S X,U S Y,V (dim C ) n − d nU d nV d nW N WU,V · W ⊠ n ⊠ ¯ W ⊠ n = X U,V,W d X d Y (dim C ) n − d nW S X,U S Y,V N WU,V · W ⊠ n ⊠ ¯ W ⊠ n = X W d X (dim C ) n − d nW δ X,Y S W,X · W ⊠ n ⊠ ¯ W ⊠ n = δ X,Y f g,X . (cid:3) heorem 4.5. Let C be a modular tensor category. Consider a spherical Z /n Z -crossedbraided permutation extension C ≀ Z /n Z = M g ∈ Z /n Z ( C ≀ Z /n Z ) g of C ⊠ n . Then Irr((
C ≀ Z /n Z ) g ) = { ( − g, X ) } X ∈ Irr( C ⊠ c ( g ) ) with g ∈ Z /n Z and fusion rules aregiven by N ( g + h,Z )( g,X ) , ( h,Y ) = k N ( p ); Z ′ ,...,Z ′ c ( g + h ) /p X ′ ,...,X ′ c ( g ) /p ,Y ′ ,...,Y ′ c ( h ) /p , where k = n − c ( g ) − c ( h ) − c ( g + h )2 p + 1 . Here g, h ∈ Z /n Z , X ∈ C ⊠ c ( g ) , Y ∈ C ⊠ c ( h ) , Z ∈ C ⊠ c ( g + h ) , p = gcd( c ( g ) , c ( h )) , and k N ( p ) indicate the (higher) fusion matrices (see (3) ) of C ⊠ p . Furthermore, X ′ i , Y ′ i , Z ′ i ∈ C ⊠ p with X = X ′ ⊠ · · · ⊠ X ′ c ( g ) /p , Y = Y ′ ⊠ · · · ⊠ Y ′ c ( h ) /p , and Z = Z ′ ⊠ · · · ⊠ Z ′ c ( h ) /p .Proof. By Proposition 2.2 the statement can be written in terms of S -matrices S ( p ) of C ⊠ p as N ( g + h,Z )( g,X ) , ( h,Y ) (13)= X W ∈ Irr( C ⊠ p ) c ( g ) p Y i =1 S ( p ) X ′ i ,W S ( p ) ,W c ( h ) p Y i =1 S ( p ) Y ′ i ,W S ( p ) ,W c ( g + h ) p Y i =1 S ( p ) Z ′ i ,W S ( p ) ,W p dim( C ) p S ( p ) ,W ! n − c ( g ) − c ( h ) − c ( g + h ) p . (14)We first note that we only need to prove the case p = 1. If p > Z / np Z -cyclic permutation extension of C ⊠ p and the formula is obtained from the p ′ = 1 formula by considering n ′ = n/p , C ′ = C ⊠ p , g ′ = g/p , and h ′ = h/p .Now, let g, h ∈ G such that gcd( c ( g ) , c ( h )) = 1 and let f g,X with X ∈ Irr( C ⊠ c ( g ) ) and f h,Y with Y ∈ Irr( C ⊠ c ( h ) ) minimal convolution idempotents. Then f g,X ◦ f h,Y = M W ∈ Irr C d X d Y d nW (dim C ) n − c ( g ) − c ( h ) S X,W ⊠ c ( g ) S Y,W ⊠ c ( h ) · W ⊠ n ⊠ ¯ W ⊠ n . Here we have used the fact that the only terms from f g,X and f h,Y that contribute to thecomposition are the coefficients of 1 R ⊠ ¯ R , where R ∈ Irr( C ⊠ n ) is of the form R = ( R ⊠ R ⊠ · · · ⊠ R c ( g ) ) ⊠ o ( g ) = ( R ′ ⊠ R ′ ⊠ · · · ⊠ R ′ c ( h ) ) ⊠ o ( h ) , with R i , R ′ i ∈ Irr( C ). However, since p = ( c ( g ) , c ( h )) = 1, we must have R = W ⊠ n for W ∈ Irr( C ) . , which gives the aboveexpression. ow let Z = Z ⊠ · · · ⊠ Z c ( g + h ) , so that f g + h,Z is a minimal convolution idempotent. Thenwe have( f g,X ◦ f h,Y ) ∗ f g + h,Z = M W,U ∈ Irr( C ) d X d Y d Z d nW d nU dim( C ) n − c ( g ) − c ( h ) − c ( g + h ) S X,W ⊠ c ( g ) S Y,W ⊠ c ( h ) S Z,U (1 W ⊠ n ⊠ ¯ W ⊠ n ∗ U ⊠ n ⊠ ¯ U ⊠ n )= M V X W,U ∈ Irr( C ) d X d Y d Z d nW d nc ( g + h ) U dim( C ) l S X,W ⊠ c ( g ) S Y,W ⊠ c ( h ) S Z,U d nW d o ( g + h ) U d o ( g + h ) V N VW ⊠ c ( g,h ) ,U · V ⊠ ⊠ ¯ V ⊠ , where l := 2 n − c ( g ) − c ( h ) − c ( g + h ). Comparing coefficients for V = 1, we obtain theequation X W,U ∈ Irr( C ) d X d Y d Z d nW dim( C ) n − c ( g ) − c ( h ) − c ( g + h ) S X,W ⊠ c ( g ) S Y,W ⊠ c ( h ) S Z, ¯ W c ( g + h ) = C ( g + h,Z )( g,X ) , ( h,Y ) d Z dim( C ) n − c ( g + h ) hence C ( g + h,Z )( g,X ) , ( h,Y ) = X W ∈ Irr( C ) d X d Y d Z d nW dim( C ) n − c ( g ) − c ( h ) S X,W ⊠ c ( g ) S Y,W ⊠ c ( h ) S Z, ¯ W c ( g + h ) . We now see d + f g,X = d X dim( C ) n − c ( g )2 . Hence normalizing we obtain N ( g + h,Z )( g,X ) , ( h,Y ) = X W ∈ Irr( C ) p dim( C ) n − c ( g ) − c ( h ) − c ( g + h ) d nW S X,W ⊠ c ( g ) S Y,W ⊠ c ( h ) S Z, ¯ W c ( g + h ) . The right hand side factorizes into the expression (14). (cid:3)
Note that in the case that the genus n − c ( g ) − c ( h ) − c ( g + h )2 p + 1 vanishes, we have that N ( g + h,Z )( g,X ) , ( h,Y ) = dim C ⊠ p (cid:0) X ′ ⊗ · · · ⊗ X ′ c ( g ) p ⊗ Y ′ ⊗ · · · ⊗ Y ′ c ( h ) p , Z ′ ⊗ · · · ⊗ Z ′ c ( g + h ) p (cid:1) for example, we recover a well-known special case N (0 ,Z ⊠ ··· ⊠ Z n )( g, ) , ( − g, ) = N Z ,...,Z n first observed for multiplicities in n -interval inclusions [KLM01] and later for fusion rules incyclic permutations [LX04] of conformal nets.In order for the equality with the dimension of the modular functor vector spaces to makesense, we need that the exponent n − c ( g ) − c ( h ) − c ( g + h ) p is even. As a consistency check, verifythe following lemma which implies this is indeed the case. Lemma 4.6. n − ( m,n ) − ( k,n ) − ( m + k,n )(( m,n ) , ( k,n )) is even. roof. First we claim the set of numbers { ( m,n )(( m,n ) , ( k,n )) , ( k,n )(( m,n ) , ( k,n )) , ( m + k,n )(( m,n ) , ( k,n )) } is pairwise co-prime. Note that ( k,n )(( m,n ) , ( k,n )) and ( m,n )(( m,n ) , ( k,n )) are coprime. Suppose l | ( m + k,n )(( m,n ) , ( k,n )) and l | ( m,n )(( m,n ) , ( k,n )) .Then we must also have l | ( k,n )(( m,n ) , ( k,n )) , but since this is coprime to l | ( m,n )(( m,n ) , ( k,n )) , it must be 1.A similar argument applies switching m and k , and we get that these numbers are coprime.We break the rest of the proof into parity cases.(1) Suppose n (( m,n ) , ( k,n )) is odd. Then since the other three terms in the expression mustdivide this one, they are also odd so we see that the whole expression is even (odd-odd-odd-odd=even).(2) Suppose n (( m,n ) , ( k,n )) is even and at least one of the other three terms is even. Thenthe other 2 must be odd since they are pairwise co-prime. Thus the whole expressionis even, since (even-even-odd-odd)=even.(3) Suppose n (( m,n ) , ( k,n )) is even, but the other three terms are all odd. We claim this is notpossible. Indeed, suppose ( m,n )(( m,n ) , ( k,n )) , ( k,n )(( m,n ) , ( k,n )) are both odd. Then m (( m,n ) , ( k,n )) , k (( m,n ) , ( k,n )) must both be odd hence m + k ((( m,n ) , ( k,n )) must be even, thus 2 | m + k ((( m,n ) , ( k,n )) and 2 | n ((( m,n ) , ( k,n )) by hypothesis, so ( m + k,n )((( m,n ) , ( k,n )) is even. Thus this cannot occur. (cid:3) Remark 4.7.
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The fusion coefficients g N τ n are given by the Fibonacci and Lucasnumbers for g = 0 ,
1, respectively.( i, )(2 , ) = 2( i + 2 , ) + ( i + 2 , τ )( i, )(2 , τ ) = ( i + 2 , ) + 3( i + 2 , τ )( i, )(2 , τ τ ) = 3( i + 2 , ) + 4( i + 2 , τ )( i, τ )(2 , ) = ( i + 2 , ) + 3( i + 2 , τ )( i, τ )(2 , τ ) = 3( i + 2 , τ ) + 4( i + 2 , τ )( i, τ )(2 , τ τ ) = 4( i + 2 , τ ) + 7( i + 2 , τ )(2 , )(2 , ) = + τ τ + τ τ + τ τ τ τ (2 , )(2 , τ ) = τ + τ + τ τ + τ τ τ + τ τ τ + τ τ τ τ (2 , )(2 , τ τ ) = + τ + · · · + τ τ + · · · + τ τ τ + · · · + τ τ τ τ (2 , τ )(2 , τ ) = + τ + τ + 2 τ τ + τ τ + τ τ τ + τ τ τ + 2 τ τ τ τ (2 , τ )(2 , τ τ ) = τ + τ + τ τ + τ τ + τ τ + τ τ + τ τ ++ 2 τ τ τ + 2 τ τ τ + τ τ τ + τ τ τ + 2 τ τ τ τ (2 , τ τ )(2 , τ τ ) = + τ + · · · + τ τ + 2 τ τ + · · · + 2 τ τ τ + · · · + 4 τ τ τ τ where we write “ · · · ” for obvious permutation of objects. Department of Mathematics, Ohio University, Athens, OH 45701, USA
E-mail address : [email protected] E-mail address : [email protected] Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
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