Algebraic structures in group-theoretical fusion categories
Yiby Morales, Monique Müller, Julia Plavnik, Ana Ros Camacho, Angela Tabiri, Chelsea Walton
aa r X i v : . [ m a t h . QA ] J a n ALGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSIONCATEGORIES
YIBY MORALES, MONIQUE M ¨ULLER, JULIA PLAVNIK, ANA ROS CAMACHO,ANGELA TABIRI, AND CHELSEA WALTON
Abstract.
It was shown by Ostrik (2003) and Natale (2017) that a collectionof twisted group algebras in a pointed fusion category serve as explicit Moritaequivalence class representatives of indecomposable, separable algebras in suchcategories. We generalize this result by constructing explicit Morita equiva-lence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the ‘free functor’Φ from a pointed fusion category to a group-theoretical fusion category with amonoidal structure. Our algebras of interest are then constructed as the imageof twisted group algebras under Φ. We also show that twisted group algebrasadmit the structure of Frobenius algebras in a pointed fusion category, andwe establish a Frobenius monoidal structure on Φ as well. As a consequence,our algebras are Frobenius algebras in a group-theoretical fusion category, andlike twisted group algebras in the pointed case, they also enjoy several goodalgebraic properties.
Contents
1. Introduction 22. Preliminaries on monoidal categories 52.1. Monoidal categories and functors 52.2. Algebraic structures in monoidal categories 72.3. Categories of modules over algebras 102.4. Categories of bimodules over algebras 112.5. On Morita equivalence of algebras 123. A Frobenius monoidal functor Φ to a category of bimodules 154. Morita equivalence of algebras in a category of bimodules 205. Algebras A ( L, ψ ) in pointed fusion categories
Vec ωG A K,β ( L, ψ ) in group-theoretical fusion categories C ( G, ω, K, β ) 287. Representation theory of group-theoretical fusion categories 36Appendix A. Remainder of the proof of Theorem 3.2 39Appendix B. Remainder of the proof of Theorem 4.1 44Appendix C. Direct proof that A K,β ( L, ψ ) is Frobenius 46References 54
Mathematics Subject Classification.
Key words and phrases.
Frobenius algebra, Frobenius monoidal functor, group-theoretical fu-sion category, Morita equivalence, pointed fusion category, separable algebra. Introduction
The goal of this work is to construct explicit algebras that represent Moritaequivalence classes in group-theoretical fusion categories, and that possess goodalgebraic properties. Throughout, we assume that k is an algebraically closed fieldof characteristic 0, and an unadorned ⊗ denotes ⊗ k .A group-theoretical fusion category is a certain kind of monoidal category whoseconstruction depends on group-theoretic data, and we will restrict our attentionto such categories below. But for now let us discuss the prevalence of Moritaequivalence of algebras in monoidal categories in general. Recall that two rings aresaid to be Morita equivalent if their categories of modules are equivalent as abeliancategories. Many nice properties are preserved under such an equivalence includingthe Noetherian, (semi)simple, (semi)hereditary, and (semi)prime conditions [24,Chapter 7]. The notion of Morita equivalence has been upgraded for algebrasof various types, and is used in several areas including C ∗ -algebras [3], Poissongeometry [34], and various subfields of physics [7, 15, 31]; we will discuss [15]below. In all of these cases, one is studying the Morita equivalence of algebras(or, of algebra objects) in a fixed monoidal category. We provide a review of thiscategorical terminology, including the definition of Morita equivalence and of specialtypes of algebras under consideration, in Section 2.A special use of Morita equivalence occurs in two-dimensional rational confor-mal field theories (rCFTs). These are certain quantum field theories that displayconformal symmetry, and they have inspired vital mathematical structures such asvertex operator algebras [2] and modular tensor categories (MTCs) [1, Chapter 3].Often, algebras in MTCs provide a useful way of classifying and describing thephysical quantities in rCFTs. In particular, full rCFTs [15] are completely fixed bypairs ( ν, A ), where ν is a rational vertex operator algebra encoding the symmetriesof the rCFT, and A is a separable, symmetric Frobenius algebra in Rep ( ν ) (whichis an MTC [21]). This algebra A is the algebra of boundary fields associated toone given boundary condition of the full rCFT. Moreover, the algebras arising fromboundary conditions of the full rCFT are all Morita equivalent. So, essentially, thecollection of full rCFTs are in bijection with Morita equivalence classes of separable,symmetric Frobenius algebras in MTCs of the form Rep ( ν ).Now returning to the goal of our work, we discuss how the aim is resolvedpartially for an arbitrary fusion category C by work of V. Ostrik [30]. Two algebrasin C are said to be Morita equivalent if their categories of (right) modules in C areequivalent as (left) C -module categories; see Section 2.3. The main result of [30]states that any C -module category M is equivalent to the category of modules oversome algebra A in C , and the algebra A used in the proof of this result is an internalEnd of any nonzero object of M (see [30, Section 3.2]). It is also shown that thisinternal End A can be taken to be connected [Definition 2.10], but no other goodalgebraic properties of A are established nor is the construction of A explicit. Incontrast, we restrict our attention to certain types of fusion categories that dependon group-theoretic data and, using a construction different than internal Ends, LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 3 we produce Morita equivalence representatives of algebras in these categories thatdepend explicitly on this group-theoretic data.Such a strategy was used to resolve the goal in the setting of pointed fusioncategories, that is, for the categories
Vec ωG , with G a finite group and ω ∈ H ( G, k × ),consisting of G -graded k -vector spaces with associativity constraint ω . The simpleobjects of Vec ωG are 1-dimensional k -vector spaces, denoted by { δ g } g ∈ G , with G -grading ( δ g ) x = δ g,x k , for g, x ∈ G . Indeed, we have the following constructionand result due to work of V. Ostrik and work of S. Natale. Definition 1.1.
Let L be a subgroup of G so that ω | L × is trivial, and take a2-cochain ψ ∈ C ( L, k × ) so that dψ = ω | L × . The twisted group algebra A ( L, ψ ) in
Vec ωG is L g ∈ L δ g , with multiplication given by δ g ⊗ δ g ′ ψ ( g, g ′ ) δ gg ′ . Theorem 1.2. [29, Example 2.1] [8, Example 9.7.2] [28]
A collection of twistedgroup algebras A ( L, ψ ) serve as Morita equivalence class representatives of inde-composable, separable algebras in the pointed fusion category Vec ωG . (cid:3) The first of our results is that we establish a Frobenius algebra structure on thetwisted group algebras and study related algebraic properties. See Definition 2.10for a description of the properties mentioned below.
Proposition 1.3 (Propositions 5.7 and 5.9) . The twisted group algebras A ( L, ψ ) admit the structure of a Frobenius algebra in Vec ωG . They are also indecomposableand separable in Vec ωG , are connected, are special, and are symmetric if and only if ω ( g − , g, g − ) = 1 for each g ∈ L . (cid:3) Now we turn our attention to group-theoretical fusion categories . Introduced byP. Etingof, D. Nikshych, and V. Ostrik in [10, Section 8.8], these are the categories C ( G, ω, K, β ) consisting of A ( K, β )-bimodules in
Vec ωG , for G and ω as above, andwith K a subgroup of G so that ω | K × is trivial, and β ∈ C ( K, k × ) so that dβ = ω | K × . (See also [8, Section 9.7].) Group-theoretical fusion categories are avital part of the classification program of general fusion categories (see, e.g., [11,Theorem 9.2] and [8, Section 9.13]), and due to their explicit construction, theyalso serve as a go-to testing ground for results about fusion categories (see, e.g., [9,Section 5], [13, Corollary 4.4], [18], [19, Section 4], [27], [29]).Towards our goal of constructing nice Morita equivalence class representativesof algebras in group-theoretical fusion categories, we start in a more general set-ting and consider the ‘free’ functor from a monoidal category C to a category ofbimodules in C , and impose on this functor further structure (see Definition 2.2). Theorem 1.4 (Theorem 3.2) . Let ( C , ⊗ ) be a monoidal category, and let A be aspecial Frobenius algebra in C . Let A C A denote the monoidal category of A -bimodulesin C . Then, the functor Φ :
C → A C A , which sends objects X to ( A ⊗ X ) ⊗ A , andmorphisms ϕ to (id A ⊗ ϕ ) ⊗ id A , admits the structure of a Frobenius monoidalfunctor. (cid:3) The result above enables us to define algebraic structures that will fulfill our goal.
MORALES ET AL.
Definition-Theorem 1.5 (Definition 6.3, Theorem 6.4) . Using the free functor Φ above in the case when C = Vec ωG and A = A ( K, β ) , we define the twisted Heckealgebra A K,β ( L, ψ ) to be the algebra Φ( A ( L, ψ )) in C ( G, ω, K, β ) . It admits thestructure of a Frobenius algebra in C ( G, ω, K, β ) , an explicit description of which isknown. (cid:3) The terminology is due to the fact that simple objects of group-theoretical fusioncategories C ( G, ω, K, β ) are in part parameterized by K -double cosets in G (seeLemma 6.2), and the multiplication of A K,β ( L, ψ ) is twisted by cocycles β and ψ .Twisted Hecke algebras also enjoy several nice algebraic properties. Proposition 1.6 (Proposition 6.10) . The twisted Hecke algebras A K,β ( L, ψ ) areindecomposable and separable (Frobenius algebras) in C ( G, ω, K, β ) , and are alsospecial. (cid:3) We provide a precise condition describing when twisted Hecke algebras are con-nected in Proposition 6.13; in general, the connected property does not hold. Inany case, the twisted Hecke algebras in C ( G, ω, K, β ) are nearly as nice algebraicallyas the twisted group algebras in
Vec ωG ; cf. Proposition 1.3. We inquire about thesymmetric property for twisted Hecke algebras in Question 6.11, which involvesunderstanding the explicit rigidity structure of group-theoretical fusion categories(see Question 2.22); this is reserved for future work.Finally, our goal is achieved as follows. Theorem 1.7 (Theorem 7.4) . A collection of twisted Hecke algebras A K,β ( L, ψ ) serve as Morita equivalence class representatives of indecomposable, separable alge-bras in the group-theoretical fusion category C ( G, ω, K, β ) . (cid:3) An application of this result to P. Etingof, R. Kinser, and the last author’sstudy of tensor algebras in group-theoretical fusion categories [9] is discussed inRemark 7.6 and Example 7.7.Theorem 1.7 is achieved by introducing the notion of a
Morita preservingmonoidal functor [Theorem 4.1, Definition 4.3] and by establishing the followinggeneral result.
Theorem 1.8 (Theorem 4.9) . Let C be a rigid monoidal category. Take a specialFrobenius algebra A in C , and take algebras B , B ′ in C . Recall the monoidal functor Φ from Theorem 1.4. Then, B and B ′ are Morita equivalent as algebras in C if andonly if Φ( B ) and Φ( B ′ ) are Morita equivalent as algebras in A C A . (cid:3) Indeed, with Theorem 1.2 (due to Ostrik and Natale) and Proposition 1.3, The-orem 1.8 provides the crucial step in proving Theorem 1.7 by setting C = Vec ωG , A = A ( K, β ), B = A ( L, ψ ), B ′ = A ( L ′ , ψ ′ ). Acknowledgements.
The authors thank C´esar Galindo and Ryan Kinser forinsightful comments on a preliminary version of this article. This work began atthe Women in Noncommutative Algebra and Representation Theory (WINART2)
LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 5 workshop, held at the University of Leeds in May 2019. We thank the Universityof Leeds’ administration and staff for their hospitality and productive atmosphere.Y. Morales was partially supported by the London Mathematical Society, work-shop grant
Preliminaries on monoidal categories
In this section, we provide background information and preliminary results onmonoidal categories and monoidal functors in Section 2.1, on algebraic structures inmonoidal categories in Section 2.2, and on categories of modules and bimodules overalgebras in monoidal categories in Sections 2.3 and 2.4. In Section 2.5, we establishpreliminary results on Morita equivalence of algebras in monoidal categories.To begin, take an abelian category C with a zero object 0 throughout this article.By X ∈ C we mean that X is an object of C . A nonzero X ∈ C is simple if 0 and X are its only subobjects. A category C is semisimple if every object is a directsum of simple objects. We say that X ∈ C is indecomposable if it is nonzero andcannot be decomposed as the direct sum of nonzero subobjects. Simple objects in C are indecomposable; the converse holds when C is semisimple. We assume that allcategories in this work are locally small , i.e., for any objects X, Y ∈ C the collectionof morphisms from X to Y is a set.Moreover, for a field k , a k -linear category C is locally finite if each Hom spaceis a finite-dimensional k -vector space and if every object has finite length. Wealso say that a k -linear category C is finite if it is equivalent to the category offinite-dimensional modules over some finite-dimensional k -algebra.2.1. Monoidal categories and functors.Definition 2.1. (see, e.g., [8, Definition 2.2.8]) A monoidal category C consists ofthe following data: • a category C , • a bifunctor ⊗ : C × C → C , • an object ∈ C , • a natural isomorphism α X,X ′ ,X ′′ : ( X ⊗ X ′ ) ⊗ X ′′ ∼ → X ⊗ ( X ′ ⊗ X ′′ ) foreach X, X ′ , X ′′ ∈ C , • natural isomorphisms l X : ⊗ X ∼ → X, r X : X ⊗ ∼ → X for each X ∈ C ,such that the pentagon and triangle axioms are satisfied [8, (2.2),(2.10)]. MORALES ET AL.
Definition 2.2. [32, page 85] [6] [33, (6.46), (6.47)]Let ( C , ⊗ C , C , α ∗ , ∗ , ∗ , l ∗ , r ∗ ) and ( D , ⊗ D , D , α ∗ , ∗ , ∗ , l ∗ , r ∗ ) be monoidal categories.(a) A monoidal functor ( F, F ∗ , ∗ , F ) : C → D consists of the following data: • a functor F : C → D , • a natural transformation F X,X ′ : F ( X ) ⊗ D F ( X ′ ) → F ( X ⊗ C X ′ ) forall X, X ′ ∈ C , • a morphism F : D → F ( C ) in D ,that satisfy the following associativity and unitality constraints, F X,X ′ ⊗ C X ′′ (id F ( X ) ⊗ D F X ′ ,X ′′ ) α F ( X ) ,F ( X ′ ) ,F ( X ′′ ) = F ( α X,X ′ ,X ′′ ) F X ⊗ C X ′ ,X ′′ ( F X,X ′ ⊗ D id F ( X ′′ ) ) ,F ( l X ) − l F ( X ) = F C ,X ( F ⊗ D id F ( X ) ) ,F ( r X ) − r F ( X ) = F X, C (id F ( X ) ⊗ D F ) . (b) A comonoidal functor ( F, F ∗ , ∗ , F ) : C → D consists of the following data: • a functor F : C → D , • a natural transformation F X,X ′ : F ( X ⊗ C X ′ ) → F ( X ) ⊗ D F ( X ′ ) forall X, X ′ ∈ C , • a morphism F : F ( C ) → D in D ,that satisfy the following coassociativity and counitality constraints, α − F ( X ) ,F ( X ′ ) ,F ( X ′′ ) (id F ( X ) ⊗ D F X ′ ,X ′′ ) F X,X ′ ⊗ C X ′′ = ( F X,X ′ ⊗ D id F ( X ′′ ) ) F X ⊗ C X ′ ,X ′′ F ( α − X,X ′ ,X ′′ ) ,F ( l X ) = l F ( X ) ( F ⊗ D id F ( X ) ) F C ,X ,F ( r X ) = r F ( X ) (id F ( X ) ⊗ D F ) F X, C . (c) A Frobenius monoidal functor ( F, F ∗ , ∗ , F , F ∗ , ∗ , F ) from C to D is a functorwhere ( F, F ∗ , ∗ , F ) is monoidal and ( F, F ∗ , ∗ , F ) is comonoidal, such thatfor all X, X ′ , X ′′ ∈ C :( F X,X ′ ⊗ D id F ( X ′′ ) ) α − F ( X ) ,F ( X ′ ) ,F ( X ′′ ) (id F ( X ) ⊗ D F X ′ ,X ′′ )= F X ⊗ C X ′ ,X ′′ F ( α − X,X ′ ,X ′′ ) F X,X ′ ⊗ C X ′′ , (id F ( X ) ⊗ D F X ′ ,X ′′ ) α F ( X ) ,F ( X ′ ) ,F ( X ′′ ) ( F X,X ′ ⊗ D id F ( X ′′ ) )= F X,X ′ ⊗ C X ′′ F ( α X,X ′ ,X ′′ ) F X ⊗ C X ′ ,X ′′ . Here, ‘monoidal’ means ‘lax monoidal’ in other references. Strong monoidalfunctors are monoidal functors where F ∗ , ∗ are F are isomorphisms in D , and wedo not require this condition here. Definition 2.3. (see, e.g., [8, Sections 7.1, 7.2]) Let ( C , ⊗ C , C , α ∗ , ∗ , ∗ , l ∗ , r ∗ ) be amonoidal category.(a) A left C -module category is a category M equipped with • a bifunctor ⊗ : C × M → M , • natural isomorphisms for associativity m X,Y,M : ( X ⊗ Y ) ⊗ M ∼ → X ⊗ ( Y ⊗ M ) , ∀ X, Y ∈ C , M ∈ M satisfying the pentagon axiom, and
LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 7 • for each M ∈ M a natural isomorphism ⊗ M ∼ → M satisfying thetriangle axiom. Right C -module categories are defined analogously.(b) A module category M over C is indecomposable if it is nonzero and is notequivalent to a direct sum of two nontrivial module categories over C .(c) Let M and N be two left C -module categories. A (left) C -module functor from M to N is a functor F : M → N equipped with a natural isomorphism s X,M : F ( X ⊗ M ) ∼ → X ⊗ F ( M ) for each X ∈ C , M ∈ M satisfyingthe pentagon and triangle axioms. Right C -module functors are definedanalogously.(d) An equivalence of C -module categories is a C -module functor ( F, s ) so that F : M → N is an equivalence of categories.Now we recall terminology for monoidal categories with dual objects.
Definition 2.4.
Let C be a monoidal category. An object in C is called rigid ifit has left and right duals. Namely, for each X ∈ C , there exists objects X ∗ and ∗ X ∈ C so that we have co/evaluation maps ev X : X ∗ ⊗ X → , coev X : → X ⊗ X ∗ , ev ′ X : X ⊗ ∗ X → , coev ′ X : → ∗ X ⊗ X, satisfying compatibility conditions [8, (2.43)–(2.46)]. The monoidal category C iscalled rigid if each of its objects is rigid.Later in Sections 5, 6, and 7, we will focus our attention on k -linear categories.So, consider the following terminology. Definition 2.5. [8, Sections 2.1, 2.10 and 4.1] Let C be a k -linear, locally-finite,monoidal category. (Recall we assume that C is abelian.)(a) We call C a multi-tensor category if it is also rigid. If, in addition, End C ( ) ∼ = k (i.e., if is a simple object of C ), then C is a tensor category .(b) A multi-fusion category is a finite semisimple multi-tensor category. A fu-sion category is a multi-fusion category with End C ( ) ∼ = k , i.e., a finitesemisimple tensor category.With extra structure on C , we require more structure of its module categories.The notion below will be of use later. Definition 2.6. [30, Section 2.2, Definition 2.6] A module category over a fusioncategory C is a C -module category M as in Definition 2.3 that is, in addition,semisimple, k -linear, and abelian so that its bifunctor ⊗ : C × M → M is bilinearon morphisms and is exact.2.2.
Algebraic structures in monoidal categories.
Now we recall the notion ofan algebra, a coalgebra, and a Frobenius algebra in a monoidal category. For generalinformation, see [16, Section 2], [30, Section 3], [8, Section 7.8], and referenceswithin.
MORALES ET AL.
Definition 2.7 ( Alg ( C ), Coalg ( C ), FrobAlg ( C )) . Let ( C , ⊗ , , α, l, r ) be a monoidalcategory.(a) An algebra in C is a triple ( A, m, u ), with A ∈ C , and m : A ⊗ A → A (multiplication), u : → A (unit) being morphisms in C , satisfying unitalityand associativity constraints: m ( m ⊗ id) = m (id ⊗ m ) α A,A,A , m ( u ⊗ id) = l A , m (id ⊗ u ) = r A . A morphism of algebras ( A, m A , u A ) to ( B, m B , u B ) is a map f : A → B in C so that f m A = m B ( f ⊗ f ) and f u A = u B . Algebras in C and theirmorphisms form a category, which we denote by Alg ( C ).(b) A coalgebra in C is a triple ( C, ∆ , ε ), where C ∈ C , and ∆ : C → C ⊗ C (comultiplication) and ε : C → (counit) are morphisms in C , satisfyingcounitality and coassociativity constraints: α C,C,C (∆ ⊗ id)∆ = (id ⊗ ∆)∆ , ( ε ⊗ id)∆ = l − C , (id ⊗ ε )∆ = r − C . A morphism of coalgebras ( C, ∆ C , ε C ) to ( D, ∆ D , ε D ) is a morphism g : C → D in C so that ∆ D g = ( g ⊗ g )∆ C and ε D g = ε C . Coalgebrasin C and their morphisms form a category, which we denote by Coalg ( C ).(c) A Frobenius algebra in C is a tuple ( A, m, u, ∆ , ε ), where ( A, m, u ) ∈ Alg ( C )and ( A, ∆ , ε ) ∈ Coalg ( C ), so that( m ⊗ id) α − A,A,A (id ⊗ ∆) = ∆ m = (id ⊗ m ) α A,A,A (∆ ⊗ id) . A morphism of Frobenius algebras in C is a morphism in C that lies in both Alg ( C ) and Coalg ( C ). Frobenius algebras in C and their morphisms form acategory, which we denote by FrobAlg ( C ). Remark 2.8.
Alternatively, a Frobenius algebra in C is a tuple ( A, m, u, p, q ),where p : A ⊗ A → and q : → A ⊗ A are morphisms in C satisfying an invariancecondition, p (id A ⊗ m ) α A,A,A = p ( m ⊗ id A ), and the Snake Equation, r A (id A ⊗ p ) α A,A,A ( q ⊗ id A ) l − A = id A = l A ( p ⊗ id A ) α − A,A,A (id A ⊗ q ) r − A . In this case, we call p a non-degenerate pairing . To convert from ( A, m, u, p, q ) to(
A, m, u, ∆ , ε ) in Definition 2.7(c), take∆ := ( m ⊗ id A ) α − A,A,A (id A ⊗ q ) r − A and ε := p ( u ⊗ id A ) r − A . On the other hand, to convert from (
A, m, u, ∆ , ε ) to ( A, m, u, p, q ), take p := ε A m A and q := ∆ A u A . In fact, one can see that a Frobenius algebra in a monoidal category is a self-dualobject with evaluation and coevaluation maps given by p and q , respectively. See[17] an [22, Section 2.3] for more details. LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 9
We have another equivalent definition of a Frobenius algebra in C in the casewhen C is rigid; this is given in Remark 2.16 below.Next, we recall how the functors of Definition 2.2 preserve the algebraic struc-tures in Definition 2.7. Proposition 2.9. [32, p.100-101] [33, Lemma 2.1] [6, Corollary 5] [23, Prop. 2.13](a)
Let ( F, F ∗ , ∗ , F ) : C → D be a monoidal functor. If ( A, m, u ) ∈ Alg ( C ) , then ( F ( A ) , F ( m ) F A,A , F ( u ) F ) ∈ Alg ( D ) . (b) Let ( F, F ∗ , ∗ , F ) : C → D be a comonoidal functor. If ( C, ∆ , ε ) ∈ Coalg ( C ) ,then ( F ( C ) , F C,C F (∆) , F F ( ε )) ∈ Coalg ( D ) . (c) Let ( F, F ∗ , ∗ , F , F ∗ , ∗ , F ) : C → D be a Frobenius monoidal functor. If ( A, m, u, ∆ , ε ) ∈ FrobAlg ( C ) , then ( F ( A ) , F ( m ) F A,A , F ( u ) F , F A,A F (∆) , F F ( ε )) ∈ FrobAlg ( D ) . (cid:3) Some properties of the structures in Definition 2.7 of interest here are givenbelow.
Definition 2.10.
Take ( C , ⊗ , , α, l, r ) a k -linear, monoidal category.(a) A ∈ Alg ( C ) is indecomposable if it is not isomorphic to a direct sum ofnonzero algebras in C .(b) A ∈ Alg ( C ) is connected (or haploid ) if dim k Hom C ( , A ) = 1, that is, if theunit of A is unique up to scalar multiple.(c) A ∈ Alg ( C ) is separable if there exists a morphism ∆ ′ : A → A ⊗ A in C sothat m ∆ ′ = id A as maps in C with(id A ⊗ m ) α A,A,A (∆ ′ ⊗ id A ) = ∆ ′ m = ( m ⊗ id A ) α − A,A,A (id A ⊗ ∆ ′ ) . (d) ( A, m, u, ∆ , ε ) ∈ FrobAlg ( C ) is special if m ∆ = id A and εu = ϕ id for anonzero ϕ ∈ k . Remark 2.11. (a) We have that an algebra A in C is indecomposable if andonly if it is not the direct sum of proper subalgebras of A . Here, a subalgebra of A is, by definition, a subobject of A with multiplication and unit inducedby those of A , and it is proper if it is not equal to the zero object or A itself.(b) It is clear from the definitions that a connected algebra must be indecom-posable.(c) The displayed equations in Definition 2.10(c) above are the requirementthat ∆ ′ is both a left and a right A -module map, so that m splits as a mapof A -bimodules in C ; see Sections 2.3 and 2.4.(d) The special Frobenius condition above implies separability.In order to ask for a Frobenius algebra to be symmetric in the categorical sense,we work in a rigid monoidal category; see [15, 17]. Definition 2.12.
Let ( C , ⊗ , , α, l, r, ∗ () , () ∗ ) be a rigid monoidal category. We saythat a Frobenius algebra ( A, m, u, ∆ , ε ) is symmetric if ∗ A = A ∗ as objects in C ,and if εm is equal to Ω A := ev A (id A ∗ ⊗ l A ) (id ∗ A = A ∗ ⊗ εm ⊗ id A ) ( α ∗ A,A,A ⊗ id A ) (( coev ′ A ⊗ id A ) l − A ⊗ id A ) as morphisms in C . Example 2.13.
Suppose that C = Vec k , the category of finite-dimensional k -vector spaces. Then the definition above recovers the usual notion of a symmetricFrobenius k -algebra. Indeed, for A a Frobenius k -algebra and x, y ∈ A we get thatΩ A ( x ⊗ y ) = ev A (id A ∗ ⊗ l A )(id ∗ A = A ∗ ⊗ εm ⊗ id A )( P i ( ∗ e i ⊗ ( e i ⊗ x )) ⊗ y )= ev A (id A ∗ ⊗ l A )( P i ( e ∗ i ⊗ εm ( e i ⊗ x )) ⊗ y )= ev A ( P i εm ( e i ⊗ x )( e ∗ i ⊗ y )= εm ( y ⊗ x ) . Categories of modules over algebras.
Fix C := ( C , ⊗ , , α, l, r ) a monoidalcategory. Now we turn our attention to modules over algebras in C . For moredetails, see [30, Section 3] and [8, Section 7.8]. Definition 2.14 ( ρ M , ρ AM , λ M , λ AM , C A , A C ) . Take A := ( A, m A , u A ), an alge-bra in C . A right A -module in C is a pair ( M, ρ M ), where M ∈ C , and ρ M := ρ AM : M ⊗ A → M is a morphism in C so that ρ M ( ρ M ⊗ id A ) = ρ M (id M ⊗ m A ) α M,A,A and r M = ρ M (id M ⊗ u A ) . A morphism of right A -modules in C is a morphism f : M → N in C so that f ρ M = ρ N ( f ⊗ id A ). Right A -modules in C and their morphisms form a category, which wedenote by C A . The category A C of left A -modules ( M, λ M := λ AM : A ⊗ M → M ) in C is defined likewise.We have that C A is a left C -module category: for X ∈ C and ( M, ρ M ) ∈ C A , thebifunctor C × C A → C A is defined by( X ⊗ M ) ⊗ A α X,M,A −−−−−−−−−−→ X ⊗ ( M ⊗ A ) id X ⊗ ρ M −−−−−−−−−−→ X ⊗ M. Similarly, A C is a right C -module category. Proposition 2.15. [30, Remark 3.1] [8, Proposition 7.8.30]
We have that C A isan indecomposable (resp., semisimple) C -module category if A is an indecomposable(resp., separable) algebra in C . (cid:3) We have an equivalent definition of a Frobenius algebra in a (rigid) monoidalcategory, which uses the terminology above.
Remark 2.16.
Let C be a rigid monoidal category. Then ∗ A is a left A -module via λ ∗ A := (id ∗ A ⊗ ev ′ A ) α ∗ A,A, ∗ A (id ∗ A ⊗ m A ⊗ id ∗ A )( α ∗ A,A,A ⊗ id ∗ A )( coev ′ A ⊗ id A ∗ A )( l − A ⊗ id A ∗ A ) . Now by [17], a Frobenius algebra in C can be equivalently defined as an algebra A in C so that ( A, λ A ) is isomorphic to ( ∗ A, λ ∗ A ) as left A -modules. LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 11
Next, we turn our attention to Morita equivalence of algebras in monoidal cate-gories.
Definition 2.17.
We say that two algebras A and B in C are Morita equivalent(in C ) if C A ∼ C B as (left) C -module categories.Several algebraic properties are preserved under Morita equivalence, such asindecomposability and separability. We will discuss a characterization of Moritaequivalence in terms of bimodules in Section 2.5.2.4. Categories of bimodules over algebras.
Fix a monoidal category C := ( C , ⊗ , , α, l, r ). We recall here preliminary notions on bimodules over al-gebras in C . For general information, see [25, Section 3.3] and [8, Section 7.8]. Definition 2.18 ( A C A ) . Take A := ( A, m A , u A ) ∈ Alg ( C ). An A -bimodule in C isa triple ( M, λ M , ρ M ), where M ∈ C , and λ M : A ⊗ M → M and ρ M : M ⊗ A → M are morphisms in C , so that ( M, λ M ) ∈ A C and ( M, ρ M ) ∈ C A with λ M (id A ⊗ ρ M ) α A,M,A = ρ M ( λ M ⊗ id A ) . A morphism of A -bimodules in C is a morphism f : M → N in C that is simultane-ously a morphism in both A C and C A . Bimodules over A in C and their morphismsform a category, which we denote by A C A . Definition 2.19 ( ⊗ A , π M,N , π
AM,N ) . Take A -bimodules M and N in C . The tensorproduct of M and N over A is the object of A C A given by M ⊗ A N := coker (cid:0) ρ M ⊗ id N − (id M ⊗ λ N ) α M,A,N (cid:1) . Let π M,N := π AM,N : M ⊗ N → M ⊗ A N denote the canonical projection, a morphismin C . Moreover, M ⊗ A N is an A -bimodule via morphisms: λ M ⊗ A N : A ⊗ ( M ⊗ A N ) → M ⊗ A N and ρ M ⊗ A N : ( M ⊗ A N ) ⊗ A → M ⊗ A N so that λ M ⊗ A N (id A ⊗ π M,N ) = π M,N ( λ M ⊗ id N ) α − A,M,N ,π M,N (id M ⊗ ρ N ) α M,N,A = ρ M ⊗ A N ( π M,N ⊗ id A ) . Proposition 2.20 (( A C A , ⊗ A , A, α A ∗ , ∗ , ∗ , l A ∗ , r A ∗ )) . [25, Section 3.3.2] The category A C A has the structure of a monoidal category with • tensor product ⊗ A , • unit object A , and • associativity constraint α AX,X ′ ,X ′′ : ( X ⊗ A X ′ ) ⊗ A X ′′ ∼ → X ⊗ A ( X ′ ⊗ A X ′′ ) for X, X ′ , X ′′ ∈ C , so that α AX,X ′ ,X ′′ π X ⊗ A X ′ ,X ′′ ( π X,X ′ ⊗ id X ′′ ) = π X,X ′ ⊗ A X ′′ (id X ⊗ π X ′ ,X ′′ ) α X,X ′ ,X ′′ , • unit constraints l AX : A ⊗ A X ∼ → X and r AX : X ⊗ A A ∼ → X so that l AX π A,X = λ X and r AX π X,A = ρ X . (cid:3) Moreover, for maps f : X → W and g : Y → Z in A C A , we get that(2.21) ( f ⊗ A g ) π X,Y = π W,Z ( f ⊗ g )as maps in C .On the other hand, [35, Section 5] and [12] discuss conditions on A to yield thatthe category of bimodules A C A above is rigid. We ask, in general: Question 2.22.
Given a (rigid) monoidal category C , what precise conditions on A ∈ Alg ( C ) need to be imposed to get that the category of bimodules A C A is rigid?2.5. On Morita equivalence of algebras.
We provide here characterizationsfor the Morita equivalence of algebras in monoidal categories [Definition 2.17], andprovide other preliminary results that we will need later in Section 4. First, considerthe following notation.
Definition 2.23 ( α ∗ , ∗ , ∗ ) . Let C := ( C , ⊗ , , α, l, r ) be a monoidal category, andtake two algebras A and B in C . Let X, Z ∈ A C B and Y ∈ B C A . Take α X,Y,Z : ( X ⊗ B Y ) ⊗ A Z → X ⊗ B ( Y ⊗ A Z )to be the morphism in C defined by the commutative diagram: ( X ⊗ Y ) ⊗ Z π BX,Y ⊗ id Z / / α (cid:15) (cid:15) ( X ⊗ B Y ) ⊗ Z π AXY,Z / / ( X ⊗ B Y ) ⊗ A Z α (cid:15) (cid:15) ✤✤ X ⊗ ( Y ⊗ Z ) π BX,Y Z / / X ⊗ B ( Y ⊗ Z ) id X ⊗ B π AY,Z / / X ⊗ B ( Y ⊗ A Z ) . The same notation will apply in the case when the roles A and B are reversed. Lemma 2.24. [8, Exercise 7.8.28]
The morphism α exists, and is an isomorphismin C . (cid:3) Proposition 2.25.
Let C := ( C , ⊗ , , α, l, r ) be a monoidal category, and take twoalgebras A and B in C . Then the following statements hold. (a) A and B are Morita equivalent if and only if there exist bimodules P ∈ A C B and Q ∈ B C A so that P ⊗ B Q ∼ = A in A C A and Q ⊗ A P ∼ = B in B C B . (b) If there exist bimodules P ∈ A C B and Q ∈ B C A along with epimorphisms τ : P ⊗ B Q ։ A in A C A and µ : Q ⊗ A P ։ B in B C B so that the diagrams ( ∗ ) and ( ∗∗ ) below commute in C , then the equivalentconditions of part (a) hold. ( P ⊗ B Q ) ⊗ A P α / / τ ⊗ A id P (cid:15) (cid:15) (cid:15) (cid:15) P ⊗ B ( Q ⊗ A P ) id P ⊗ B µ (cid:15) (cid:15) (cid:15) (cid:15) A ⊗ A P l AP & & ▲▲▲▲▲▲▲▲▲ ( ∗ ) P ⊗ B B r BP x x rrrrrrrrr P ( Q ⊗ A P ) ⊗ B Q α / / µ ⊗ B id Q (cid:15) (cid:15) (cid:15) (cid:15) Q ⊗ A ( P ⊗ B Q ) id Q ⊗ A τ (cid:15) (cid:15) (cid:15) (cid:15) B ⊗ B Q l BQ & & ▼▼▼▼▼▼▼▼▼ ( ∗∗ ) Q ⊗ A A r AQ x x qqqqqqqqq Q LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 13
Proof. (a) This is well-known; see, e.g., [30, Remark 3.2] and [14].(b) Since C is assumed to be abelian, the category A C A is also abelian (see, e.g.,[8, Exercise 7.8.7]). So it suffices to show τ and µ are monomorphisms in A C A as epicmonomorphisms are isomorphisms in abelian categories. We prove the statementfor τ ; the proof for µ will follow similarly.Take morphisms g , g : W → P ⊗ B Q in A C A so that τ g = τ g as morphisms W → A in A C A . Consider the following commutative diagram in C , where wesuppress the ⊗ ∗ symbol in morphisms. We also invoke Lemma 2.24 in all of thediagrams below for the existence of the morphism α . ( W ⊗ A P ) ⊗ B Q α / / gi id id (cid:15) (cid:15) W ⊗ A ( P ⊗ B Q ) id τ / / gi id id (cid:15) (cid:15) W ⊗ A A gi id (cid:15) (cid:15) ( P ⊗ B Q ) ⊗ A ( P ⊗ B Q ) α (cid:15) (cid:15) id id τ ! ! ❈❈❈❈❈❈❈❈❈❈❈❈❈❈ ( P ⊗ B ( Q ⊗ A P )) ⊗ B Q α ( ( ◗◗◗◗◗◗◗◗ id µ id (cid:15) (cid:15) P ⊗ B ( Q ⊗ A ( P ⊗ B Q )) id α − u u ❥❥❥❥❥❥❥❥❥ (( P ⊗ B Q ) ⊗ A P ) ⊗ B Q α id ♥♥♥♥♥♥♥ τ id id (cid:15) (cid:15) P ⊗ B (( Q ⊗ A P ) ⊗ B Q ) id α ) ) ❚❚❚❚❚❚❚❚❚ id µ id (cid:15) (cid:15) ( P ⊗ B Q ) ⊗ A A α (cid:15) (cid:15) ( ∗ ) ( P ⊗ B B ) ⊗ B Q rBP id (cid:25) (cid:25) ✷✷✷✷✷✷✷✷✷✷ α ( ( ◗◗◗◗◗◗◗◗ P ⊗ B ( Q ⊗ A ( P ⊗ B Q )) id id τ ( ( ◗◗◗◗◗◗◗◗ ( A ⊗ A P ) ⊗ B Q lAP id * * ❱❱❱❱❱❱❱❱❱❱❱❱ P ⊗ B ( B ⊗ B Q ) id lBQ | | ③③③③③ ( ∗∗ ) P ⊗ B ( Q ⊗ A A ) id rAQ q q ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ P ⊗ B Q Now with the diagram commuting, the assumption τ g = τ g implies that(id P ⊗ B r AQ ) α P,Q,A ( g ⊗ A id A )(id W ⊗ A τ ) α W,P,Q = (id P ⊗ B r AQ ) α P,Q,A ( g ⊗ A id A )(id W ⊗ A τ ) α W,P,Q . Since α is an epimorphism by Lemma 2.24, and (id W ⊗ A τ ) is also an epimorphismby assumption, we get that(id P ⊗ B r AQ ) α P,Q,A ( g ⊗ A id A ) = (id P ⊗ B r AQ ) α P,Q,A ( g ⊗ A id A ) . By [8, Exercise 7.8.22] we have that r AQ is an isomorphism in B C , so id P ⊗ B r AQ isan isomorphism in C as well. Therefore, α P,Q,A ( g ⊗ A id A ) = α P,Q,A ( g ⊗ A id A ) . Finally, by Lemma 2.24, α is an isomorphism. Thus, ( g ⊗ A id A ) = ( g ⊗ A id A ),and g = g , as desired. (cid:3) Part (a) is a generalization of a classical ring theory result, which is presented,e.g., in [5, Theorem 4.4.5]. The proof of (b) is a generalization of [5, Lemma 4.5.2].Moreover, the result below generalizes the classic result that a k -algebra R is Moritaequivalent to a matrix algebra Mat n ( R ) over k . (Indeed, the classical result isrecovered from the following result by letting C be the monoidal category of finite-dimensional k -vector spaces with S = R and V = k ⊕ n .) Proposition 2.26.
Let C be a monoidal category, and take an algebra S in C andan object V in C so that it has a right dual in C . Then, (a) ( ∗ V ⊗ S ) ⊗ V ∈ Alg ( C ) with m ( ∗ V ⊗ S ) ⊗ V = (id ∗ V ⊗ m S ⊗ id V ) α ( r ∗ V ⊗ S ⊗ id SV )(id ∗ V S ⊗ ev ′ V ⊗ id SV ) α , for α = α − ∗ V S,V ∗ V,SV (id ∗ V S ⊗ α − V, ∗ V,SV )(id ∗ V SV ⊗ α ∗ V,S,V ) α ∗ V S,V, ∗ V SV ,α = ( α ∗ V,S,S ⊗ id V ) α − ∗ V S,S,V , and u ( ∗ V ⊗ S ) ⊗ V = (id ∗ V ⊗ u S ⊗ id V )( r − ∗ V ⊗ id V ) coev ′ V ;(b) S and ( ∗ V ⊗ S ) ⊗ V are Morita equivalent as algebras in C .Proof. (a) The algebra axioms hold in the strict case by the following commutativediagrams. We leave the non-strict case to the reader. ∗ V SV ∗ V SV ∗ V SV m id ... + + id ... m $ $ id .. ev ′ V id ..... / / id ..... ev ′ V id .. (cid:15) (cid:15) ∗ V SSV ∗ V SV id m S id .... / / id ... ev ′ V id .. (cid:15) (cid:15) ∗ V SV ∗ V SV id .. ev ′ V id .. (cid:15) (cid:15) m z z ∗ V SV ∗ V SSV id .. ev ′ V id ... / / id .... m S id (cid:15) (cid:15) ∗ V SSSV id m S id .. / / id .. m S id (cid:15) (cid:15) ∗ V SSV id m S id (cid:15) (cid:15) ∗ V SV ∗ V SV id .. ev ′ V id .. / / m ∗ V SSV id m S id / / ∗ V SV
In particular, the bottom right square commutes due to the associativity of m S . ∗ V SV ∗ V SV id .. ev ′ V id .. (cid:15) (cid:15) m } } ∗ V V ∗ V SV id ev ′ V id ... / / id u S id .... / / ∗ V SV id u S id ... / / ∗ V SSV id m S id (cid:15) (cid:15) ∗ V SV coev ′ V id ... ♦♦♦♦♦♦♦♦♦♦♦ u id ... + + ∗ V SV
Here, the bottom triangles commute due to the rigidity and unit axioms for S . Theother unit axiom for ( ∗ V ⊗ S ) ⊗ V holds likewise.(b) Let T denote the algebra ( ∗ V ⊗ S ) ⊗ V in part (a). Let P := ∗ V ⊗ S and Q := S ⊗ V . It follows from the associativity of m S , and naturality of α ∗ , ∗ , ∗ and r ∗ , that the morphisms λ TP = (id ∗ V ⊗ m S )(id ∗ V S ⊗ r S ⊗ id S )(id ∗ V S ⊗ ev ′ V ⊗ id S ) α ρ SP = (id ∗ V ⊗ m S ) α ∗ V,S,S λ SQ = ( m S ⊗ id V ) α − S,S,V ρ TQ = ( m S ⊗ id V )( r S ⊗ id SV )(id S ⊗ ev ′ V ⊗ id SV ) α , for α = (id ∗ V ⊗ α S,V, ∗ V ⊗ id S )(id ∗ V ⊗ α − SV, ∗ V,S ) α ∗ V,SV, ∗ V S ( α ∗ V,S,V ⊗ id ∗ V S ) α = ( α S,V, ∗ V ⊗ id SV )( α − SV, ∗ V,S ⊗ id V ) α − SV, ∗ V S,V , LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 15 imply that (
P, λ TP , ρ SP ) ∈ T C S and ( Q, λ SQ , ρ TQ ) ∈ S C T . Moreover, consider themorphisms b τ = (id ∗ V ⊗ m S ⊗ id V )( α ∗ V,S,S ⊗ id V ) α − ∗ V S,S,V : P ⊗ Q → T, b µ = m S ( r S ⊗ id S )(id S ⊗ ev ′ V ⊗ id S )( α S,V, ∗ V ⊗ id S ) α − SV, ∗ V,S : Q ⊗ P → S. It follows from the associativity of m S , and naturality of α ∗ , ∗ , ∗ and r ∗ , that b τ ∈ T C T and b µ ∈ S C S . It is also clear that b τ and b µ are epimorphisms in C . Moreover, themorphisms factor through epimorphisms τ : P ⊗ S Q → T and µ : Q ⊗ T P → S ,respectively, so that b τ = τ π SP,Q and b µ = µ π TQ,P . Indeed, by the naturality of α ∗ , ∗ , ∗ and the associativity of m S , we get that b τ ( ρ SP ⊗ id Q ) = b τ (id P ⊗ λ SQ ) α P,S,Q . So theclaim for τ follows from the definition of P ⊗ S Q . Likewise, the claim for µ holds.Finally, by Proposition 2.25(b), it suffices to show that τ and µ satisfy thediagrams ( ∗ ) and ( ∗∗ ) there. We will do so for ( ∗ ) in the strict case, and thegeneral case, along with ( ∗∗ ) will hold in a similar manner. The unadorned ⊗ symbol in morphisms are suppressed below. ( P ⊗ S Q ) ⊗ T P α / / τ ⊗ T id P (cid:15) (cid:15) P ⊗ S ( Q ⊗ T P ) id P ⊗ Sµ (cid:15) (cid:15) ( P ⊗ S Q ) ⊗ P πTP ⊗ SQ,P h h ◗◗◗◗◗◗◗◗◗◗ τ id P (cid:15) (cid:15) P ⊗ S ( Q ⊗ P ) id P ⊗ SπTQ,P ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ P ⊗ ( Q ⊗ T P ) πSP,Q ⊗ T P ♠♠♠♠♠♠♠♠♠♠ id P µ (cid:15) (cid:15) P ⊗ Q ⊗ P id P b µ (cid:22) (cid:22) πSP,Q id P c c πSP,Q ⊗ P O O id P πTQ,P ; ; b τ id P (cid:11) (cid:11) id ∗ V SS ev ′ V id S { { ∗ V SSS id ∗ V mS id S (cid:15) (cid:15) id ∗ V SmS ( ( ◗◗◗◗◗◗◗◗◗◗◗ T ⊗ P id ∗ V S ev ′ V id S / / λTP ! ! ❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇ πTT,P v v ♠♠♠♠♠♠♠♠♠♠♠ ∗ V SS id ∗ V SmS (cid:15) (cid:15) id ∗ V mS (cid:22) (cid:22) P ⊗ S id ∗ V mS v v ♥♥♥♥♥♥♥♥♥♥♥ ρSP } } ⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤ πSP,S ( ( ◗◗◗◗◗◗◗◗◗◗◗ T ⊗ T P lTP , , ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ ∗ V S P ⊗ S S rSP r r ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ P All regions commute either by the definitions of the maps involved, by (2.21), orby the associativity of m S . (cid:3) We will use the techniques in this proposition in the proof of Theorem 4.9 below,and the algebra in part (a) above is mentioned in Remark 7.5 below.3.
A Frobenius monoidal functor Φ to a category of bimodules In this section, take C to be a monoidal category. Our main result in this sectionis that, when A is a special Frobenius algebra in C , we establish on the ‘free’ functorfrom C to the category of A -bimodules in C a Frobenius monoidal structure. Tobegin, consider the notation below. Notation 3.1 (˜ ∗ ) . Take A ∈ Alg ( C ), and take objects X, X ′ , W ∈ A C A . For a map f : X ⊗ A X ′ → W in A C A , let ˜ f : X ⊗ X ′ → W denote its lift in C in the sense that˜ f = f π X,X ′ .Now, we have the following result. Theorem 3.2 (Φ) . Take C := ( C , ⊗ , , α, l, r ) to be a monoidal category and let A = ( A, m, u, ∆ , ε ) be a special Frobenius algebra in C . Then the following functoris Frobenius monoidal: Φ :
C → A C A X ( A ⊗ X ) ⊗ A (as objects) ϕ (id A ⊗ ϕ ) ⊗ id A (as morphisms) . Here, the monoidal structure Φ X,X ′ is defined by the lift of e Φ X,X ′ , that is, e Φ X,X ′ = Φ X,X ′ π Φ( X ) , Φ( X ′ ) , with: e Φ X,X ′ = ( α A,X,X ′ ⊗ id A )(id AX ⊗ l X ′ ⊗ id A )(id AX ⊗ ε A m A ⊗ id X ′ ,A ) α, for α := (id AX ⊗ α − A,A,X ′ ⊗ id A )( α AX,A,AX ′ ⊗ id A ) α − AXA,AX ′ ,A , and by Φ = ( r − A ⊗ id A )∆ A . Moreover, the comonoidal structure Φ X,X ′ = π Φ( X ) , Φ( X ′ ) e Φ X,X ′ is given by e Φ X,X ′ = α ′ (id AX ⊗ ∆ A u A ⊗ id X ′ ,A )(id A,X ⊗ l − X ′ ⊗ id A )( α − A,X,X ′ ⊗ id A ) , for α ′ := α AXA,AX ′ ,A ( α − AX,A,AX ′ ⊗ id A )(id AX ⊗ α A,A,X ′ ⊗ id A ) , and by Φ = m A ( r A ⊗ id A ) . Proof.
We need to verify the following conditions:(a) Φ( X ) is an A -bimodule in C ;(b) Φ X,X ′ is well defined via e Φ X,X ′ , that is,(b.1) e Φ X,X ′ ( ρ A Φ( X ) ⊗ id Φ( X ′ ) ) = e Φ X,X ′ (id Φ( X ) ⊗ λ A Φ( X ′ ) ) α Φ( X ) ,A, Φ( X ′ ) , and(b.2) Φ X,X ′ is an A -bimodule map;(c) Φ is an A -bimodule map;(d) Φ X,X ′ is an A -bimodule map;(e) Φ is an A -bimodule map;(f) the associativity condition:Φ XX ′ ,X ′′ (Φ X,X ′ ⊗ A id Φ( X ′′ ) )= Φ( α − X,X ′ ,X ′′ ) Φ X,X ′ X ′′ (id Φ( X ) ⊗ A Φ X ′ ,X ′′ ) α A Φ( X ) , Φ( X ′ ) , Φ( X ′′ ) ;(g) the unitality conditions: l A Φ( X ) = Φ( l X ) Φ ,X (Φ ⊗ A id Φ( X ) ) ,r A Φ( X ) = Φ( r X ) Φ X, (id Φ( X ) ⊗ A Φ ); LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 17 (h) the coassociativity condition: α A Φ( X ) , Φ( X ′ ) , Φ( X ′′ ) (Φ X,X ′ ⊗ A id Φ( X ′′ ) ) Φ XX ′ ,X ′′ = (id Φ( X ) ⊗ A Φ X ′ ,X ′′ ) Φ X,X ′ X ′′ Φ( α X,X ′ ,X ′′ );(i) the counitality conditions: l A Φ( X ) (Φ ⊗ A id Φ( X ) ) Φ ,X = Φ( l X ) ,r A Φ( X ) (id Φ( X ) ⊗ A Φ ) Φ X, = Φ( r X ); and (j) the Frobenius conditions:(Φ X,X ′ ⊗ A id Φ( X ′′ ) ) ( α A Φ( X ) , Φ( X ′ ) , Φ( X ′′ ) ) − (id Φ( X ) ⊗ A Φ X ′ ,X ′′ )= Φ XX ′ ,X ′′ Φ( α − X,X ′ ,X ′′ ) Φ X,X ′ X ′′ , (id Φ( X ) ⊗ A Φ X ′ ,X ′′ ) α A Φ( X ) , Φ( X ′ ) , Φ( X ′′ ) (Φ X,X ′ ⊗ A id Φ( X ′′ ) )= Φ X,X ′ X ′′ Φ( α X,X ′ ,X ′′ ) Φ XX ′ ,X ′′ . We provide some details here, but most of the proof is provided in Appendix Aand some details will be left to the reader. Note that in the diagrams below, wewill omit the ⊗ symbol in the nodes and arrows, and also omit parentheses in thearrows, to make them more compact.(a) The right and left A -module structure of Φ( X ) = ( A ⊗ X ) ⊗ A are given by ρ A Φ( X ) := (id AX ⊗ m A ) α AX,A,A ,λ A Φ( X ) := (( m A ⊗ id X ) α − A,A,X ⊗ id A ) α − A,AX,A respectively. We check the right A -module structure and the A -bimodule compat-ibility in Propositions A.1 and A.2 in the appendix. We leave the details for theverification of the left A -module condition to the reader.(b.1) Let us see that e Φ X,X ′ ( ρ Φ( X ) ⊗ id Φ( X ′ ) ) = e Φ X,X ′ (id Φ( X ) ⊗ λ Φ( X ′ ) ) α Φ( X ) ,A, Φ( X ′ ) in the strict case, and the non-strict case is verified in Proposition A.3 in theappendix. Namely, the following diagram commutes. AXAAAX ′ A id AX m A id AX ′ A / / id AXA m A id X ′ A (cid:15) (cid:15) (1) AXAAX ′ A id AX m A id X ′ A (cid:15) (cid:15) AXAX ′ A id AX ε A id X ′ A (cid:15) (cid:15) AXAAX ′ A id AX m A id X ′ A / / AXAX ′ A ✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ✐✐✐✐✐✐✐✐✐✐✐✐✐✐ id AX ε A id X ′ A / / AXX ′ A Here, (1) commutes because m A is associative. Therefore, there is a unique mapΦ X,X ′ : Φ( X ) ⊗ A Φ( X ′ ) → Φ( X ⊗ X ′ ) such that e Φ X,X ′ = Φ X,X ′ π Φ( X ) , Φ( X ′ ) . (b.2) Let us prove that Φ X,X ′ is a right A -module map when C is strict. The prooffor the rest of the part, including the non-strict case, is left to the reader. Considerthe following diagram. (Φ( X ) ⊗ A Φ( X ′ )) A (1)(2) ρ Φ( X ) ⊗ A Φ( X ′ ) (cid:15) (cid:15) Φ X,X ′ id A / / Φ( XX ′ ) A ρ Φ( XX ′ ) (cid:15) (cid:15) (3) Φ( X )Φ( X ′ ) A π Φ( X ) , Φ( X ′ ) id A k k ❳❳❳❳❳❳❳❳❳❳❳❳ e Φ X,X ′ id A ❤❤❤❤❤❤❤❤❤❤❤ id Φ( X ) ρ Φ( X ′ ) (cid:15) (cid:15) Φ( X )Φ( X ′ ) π Φ( X ) , Φ( X ′ ) s s ❣❣❣❣❣❣❣❣❣❣❣❣❣ e Φ X,X ′ + + ❱❱❱❱❱❱❱❱❱❱❱❱ Φ( X ) ⊗ A Φ( X ′ ) Φ X,X ′ / / (4) Φ( XX ′ ) We have that (1) and (4) commute by the definition of e Φ X,X ′ , and (2) commutes bythe definition of ρ Φ( X ) ⊗ A Φ( X ′ ) . Moreover, (3) commutes via the following diagram: AXAAX ′ AA id AXAAX ′ m A (cid:15) (cid:15) id AX m A id X ′ AA / / AXAX ′ AA id AXAX ′ m A (cid:15) (cid:15) id AX ε A id X ′ AA / / AXX ′ AA id AXX ′ m A (cid:15) (cid:15) AXAAX ′ A id AX m A id X ′ A / / AXAX ′ A id AX ε A id X ′ A / / AXX ′ A ; each square commutes as a result of the maps being applied in different slots.(c) We get that Φ is a right A -module map when C is strict because A is Frobenius.The non-strict case is proved in Proposition A.4 in the appendix, and we leave therest to the reader.(d) – (h) These are discussed in Appendix A: see Propositions A.5–A.9.(i) We have that Φ satisfies the counitality condition when A is special as follows.We check the left counitality constraint for C strict, and leave the rest to the reader.In the following diagram Φ( ) ⊗ A Φ( X ) Φ ⊗ A id Φ( X ) / / A ⊗ A Φ( X ) l A Φ( X ) (cid:15) (cid:15) Φ( )Φ( X ) (1) (4)(2)Φ id Φ( X ) / / π Φ( ) , Φ( X ) j j ❚❚❚❚❚❚❚❚❚❚❚❚ A Φ( X ) (3) λ Φ( X ) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙ π A, Φ( X ) ❦❦❦❦❦❦❦❦❦❦❦ Φ( X ) Φ ,X O O e Φ ,X ❥❥❥❥❥❥❥❥❥❥❥❥❥ Φ( X ) , we get that (1) commutes by the definition of Φ ,X ; (2) commutes by (2.21) and (3)commutes from the definition of l A Φ( X ) . The diagram (4) is the following: AAAXA m A id AXA / / AAXA m A id XA (cid:15) (cid:15) AAXA id A ∆ A id XA O O m A id XA / / AXA ∆ A id XA ❦❦❦❦❦❦❦❦❦❦❦❦ ❙❙❙❙❙❙❙❙❙❙❙❙ ❙❙❙❙❙❙❙❙❙❙❙❙ AXA id A u A id XA O O ❦❦❦❦❦❦❦❦❦❦❦❦❦ ❦❦❦❦❦❦❦❦❦❦❦❦❦ AXA.
LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 19
Here, the bottom left triangle commutes by the unit axiom of A , the top regioncommutes by the Frobenius compatibility condition between m A and ∆ A , and theright triangle commutes by the assumption that A is special.(j) Let us check that one of the Frobenius conditions holds for C strict; the rest isleft to the reader. Consider the diagram below. Φ( X ) ⊗ A Φ( X ′ X ′′ ) Φ X,X ′ X ′′ / / id Φ( X ) ⊗ A Φ X ′ ,X ′′ (cid:15) (cid:15) (1)(2) (6) Φ( XX ′ X ′′ ) Φ XX ′ ,X ′′ (cid:15) (cid:15) e Φ XX ′ ,X ′′ (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ Φ( X )Φ( X ′ X ′′ ) π Φ( X ) , Φ( X ′ X ′′ ) i i ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ e Φ X,X ′ X ′′ ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ id Φ( X ) Φ X ′ X ′′ (cid:15) (cid:15) id Φ( X ) e Φ X ′ X ′′ w w (8) Φ( X )(Φ( X ′ ) ⊗ A Φ( X ′′ )) π Φ( X ) , Φ( X ′ ) ⊗ A Φ( X ′′ ) v v Φ( X ) ⊗ A (Φ( X ′ ) ⊗ A Φ( X ′′ )) α A Φ( X ) , Φ( X ′ ) , Φ( X ′′ ) − (cid:15) (cid:15) (3) Φ( X )Φ( X ′ )Φ( X ′′ ) id Φ( X ) π Φ( X ′ ) , Φ( X ′′ ) O O π Φ( X ) , Φ( X ′ ) id Φ( X ′′ ) (cid:15) (cid:15) e Φ X,X ′ id Φ( X ′′ ) / / Φ( XX ′ )Φ( X ′′ ) π Φ( XX ′ ) , Φ( X ′′ ) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ (Φ( X ) ⊗ A Φ( X ′ ))Φ( X ′′ ) Φ X,X ′ id Φ( X ′′ ) ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ π Φ( X ) ⊗ A Φ( X ′ ) , Φ( X ′′ ) u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ (Φ( X ) ⊗ A Φ( X ′ )) ⊗ A Φ( X ′′ ) Φ X,X ′ ⊗ A id Φ( X ′′ ) / / (4)(5) Φ( XX ′ ) ⊗ A Φ( X ′′ ) , (7) The diagrams (2) and (4) commute from (2.21), and (3) commutes from the defi-nition of the associativity constraint α A . Moreover, (1) and (5) commute from thedefinition of Φ ∗ , ∗ , and (6) and (7) commute from the definition of Φ ∗ , ∗ . Lastly, (8)is the following diagram: AXAAX ′ X ′′ A id AXAAX ′ u A id X ′′ A (cid:15) (cid:15) id AX m A id X ′ X ′′ A / / AXAX ′ X ′′ A id AXAX ′ u A id X ′′ A (cid:15) (cid:15) id AX ε A id X ′ X ′′ A / / AXX ′ X ′′ A id AXX ′ u A id X ′′ A (cid:15) (cid:15) AXAAX ′ AX ′′ A id AX m A id X ′ AX ′′ A / / id AXAAX ′ ∆ A id X ′′ A (cid:15) (cid:15) AXAX ′ AX ′′ A id AXAX ′ ∆ A id X ′′ A (cid:15) (cid:15) id AX ε A id X ′ AX ′′ A / / AXX ′ AX ′′ A id AXX ′ ∆ A id X ′′ A (cid:15) (cid:15) AXAAX ′ AAX ′′ A id AX m A id X ′ AAX ′′ A / / AXAX ′ AAX ′′ A id AX ε A id X ′ AAX ′′ A / / AXX ′ AAX ′′ A where each square commutes because the maps are applied in different slots. (cid:3) Remark 3.3.
In the theorem above we gave the ‘free’ functor Φ :
C → A C A thestructure of a Frobenius monoidal functor when the ground algebra A is Frobenius.(a) Observe that Φ is not strong monoidal if A = C .(b) In the proof above, we did not need the full requirement that A is special;we only used the condition that m A ∆ A = id A .(c) It is natural to consider connections to its (left or right) adjoint, the forgetfulfunctor U : A C A → C . It is discussed when U admits a Frobenius monoidalstructure in [4, Theorem 6.2]; see also [33, Lemma 6.4]. In fact, we will employ the forgetful functor U in the next section to study theMorita equivalence of algebras in A C A .4. Morita equivalence of algebras in a category of bimodules
In this section, take C to be a rigid monoidal category and take A a specialFrobenius algebra in C . Our main result is on the Morita equivalence of algebras inthe monoidal category of bimodules A C A , given in Theorem 4.9 below. To begin,consider the following result and terminology. Theorem 4.1.
Let ( S , ⊗ S ) and ( T , ⊗ T ) be monoidal categories. Take a monoidalfunctor Γ :
S → T that preserves epimorphisms and so that the natural transfor-mation Γ ∗ , ∗ of Γ is an epimorphism. If S and S ′ are Morita equivalent algebras in S , then Γ( S ) and Γ( S ′ ) are Morita equivalent algebras in T .Proof. By Proposition 2.25(a), we have bimodules(
P , λ SP : S ⊗ S P → P , ρ S ′ P : P ⊗ S S ′ → P ) ∈ S S S ′ , ( Q, λ S ′ Q : S ′ ⊗ S Q → Q, ρ SQ : Q ⊗ S S → Q ) ∈ S ′ S S , equipped with isomorphisms τ : P ⊗ S ′ Q ∼ → S in S S S and µ : Q ⊗ S P ∼ → S ′ in S ′ S S ′ . Take P := Γ( P ) , Q := Γ( Q ) . By Proposition B.1, we obtain bimodules (
P, λ Γ( S ) P , ρ Γ( S ′ ) P ) ∈ Γ( S ) T Γ( S ′ ) and( Q, λ Γ( S ′ ) Q , ρ Γ( S ) Q ) ∈ Γ( S ′ ) T Γ( S ) , where λ Γ( S ) P = Γ( λ SP ) Γ S,P : Γ( S ) ⊗ T P → P, ρ Γ( S ′ ) P = Γ( ρ S ′ P ) Γ P,S ′ : P ⊗ T Γ( S ′ ) → P,λ Γ( S ′ ) Q = Γ( λ S ′ Q ) Γ S ′ ,Q : Γ( S ′ ) ⊗ T Q → Q, ρ Γ( S ) Q = Γ( ρ SQ ) Γ Q,S : Q ⊗ T Γ( S ) → Q. Consider the morphisms, where ⊗ := ⊗ S below: b τ := Γ( τ ) Γ( π S ′ P ,Q ) Γ
P ,Q : P ⊗ T Q → Γ( P ⊗ Q ) → Γ( P ⊗ S ′ Q ) → Γ( S ) , b µ := Γ( µ ) Γ( π SQ,P ) Γ
Q,P : Q ⊗ T P → Γ( Q ⊗ P ) → Γ( Q ⊗ S P ) → Γ( S ′ ) . Both b τ and b µ are epimorphisms (in Γ( S ) T Γ( S ) and Γ( S ′ ) T Γ( S ′ ) , respectively) becausethe morphisms τ , µ , π ∗ , ∗ are each epic, the natural transformation Γ ∗ , ∗ of Γ isan epimorphism, and Γ preserves epimorphisms by assumption. Moreover, theepimorphisms b τ and b µ factor through epimorphisms τ : P ⊗ Γ( S ′ ) Q ։ Γ( S ) ∈ Γ( S ) T Γ( S ) ,µ : Q ⊗ Γ( S ) P ։ Γ( S ′ ) ∈ Γ( S ′ ) T Γ( S ′ ) , so that b τ = τ π Γ( S ′ ) P,Q , b µ = µ π Γ( S ) Q,P . (4.2)Indeed, b τ ( ρ Γ( S ′ ) P ⊗ id Q ) = b τ (id P ⊗ λ Γ( S ′ ) Q ) α P, Γ( S ′ ) ,Q , LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 21 which is verified by the commutative diagram below in the strict case. The regionscommute due to the monoidal structure of Γ and by the definitions of ρ Γ( S ′ ) P , of λ S ′ Q ,of b τ , and of P ⊗ Γ( S ′ ) Q . Here, ⊗ := ⊗ S in the diagram below. P ⊗ T Γ( S ′ ) ⊗ T Q id P ⊗T λ Γ( S ′ ) Q + + id P ⊗T Γ S ′ ,Q / / ρ Γ( S ′ ) P ⊗T id Q % % Γ P,S ′ ⊗T id Q (cid:15) (cid:15) P ⊗ T Γ( S ′ ⊗ Q ) id P ⊗T Γ( λS ′ Q ) / / Γ P,S ′⊗ Q (cid:15) (cid:15) P ⊗ T Q Γ P,Q (cid:15) (cid:15) b τ (cid:2) (cid:2) Γ( P ⊗ S ′ ) ⊗ T Q Γ P ⊗ S ′ ,Q / / Γ( ρS ′ P ) ⊗T id Q (cid:15) (cid:15) Γ( P ⊗ S ′ ⊗ Q ) Γ(id P ⊗ λS ′ Q ) / / Γ( ρS ′ P ⊗ id Q ) (cid:15) (cid:15) Γ( P ⊗ Q ) Γ( πS ′ P,Q ) (cid:15) (cid:15) Γ( P ⊗ S ′ Q ) ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ Γ( τ ) (cid:15) (cid:15) P ⊗ T Q Γ P,Q / / b τ Γ( P ⊗ Q ) Γ( πS ′ P,Q ) / / Γ( P ⊗ S ′ Q ) Γ( τ ) / / Γ( S ) So the epimorphism τ exists by Definition 2.19. Likewise, the epimorphism µ exists. Finally, τ and µ satisfy diagrams ( ∗ ) and ( ∗∗ ) in Proposition 2.25(b) byProposition B.2. Therefore, by Proposition 2.25(b), the algebras Γ( S ) and Γ( S ′ )are Morita equivalent in T . (cid:3) Definition 4.3.
We call a monoidal functor Γ :
S → T
Morita preserving if itsatisfies the conclusion of Theorem 4.1.For the rest of the section, let ( C , ⊗ ) be a rigid monoidal category, and recallthat C is assumed to be abelian and locally small. Consider the following notation. Notation 4.4 ( E := E ( A )) . Take A ∈ Alg ( C ) and denote by E := E ( A ) theinternal End object that represents the functor A C A → Set , X Hom C A ( A ⊗ A X, A );see [8, Section 7.9].
Proposition 4.5. If A is a Frobenius algebra in C , then E ( A ) is a Frobenius algebrain A C A . If, further, A is special, then E ( A ) also admits the structure of a specialFrobenius algebra.Proof. The object structure of E ( A ) follows from [8, Example 7.12.8] and the ref-erences within. In particular, E ( A ) = ∗ A ⊗ A with A -bimodule structure λ AE = ( r ∗ A ⊗ id A )(id ∗ A ⊗ ev ′ A ⊗ id A )( α ∗ A,A,A ⊗ id A )(id ∗ A ⊗ m A ⊗ id ∗ AA ) ◦ ( α ∗ A,A,A ⊗ id ∗ AA )( coev ′ A ⊗ id A ∗ AA )( l − A ⊗ id ∗ AA ) α − A, ∗ A,A and ρ AE = (id ∗ A ⊗ m A ) α ∗ A,A,A .On the other hand, consider the Frobenius algebra in C . By Theorem 3.2, wethen get that Φ( ) ∈ FrobAlg ( A C A ). Now by Remark 2.16, we have an isomorphism ξ : ∗ A ∼ → A in A C . So, we define a map χ := ξ − r A ⊗ id A : Φ( ) = ( A ⊗ ) ⊗ A −→ ∗ A ⊗ A = E. It is straight-forward to check that χ is an isomorphism of objects in A C A . SinceΦ( ) is Frobenius, E also admits the structure of a Frobenius algebra in A C A .Now suppose that A is special. Then ε A u A = ϕ id for some nonzero ϕ ∈ k . Sowe get that m Φ( ) ∆ Φ( ) = ϕ id Φ( ) in this case: indeed, by Proposition 2.9(a,b),we have m Φ( ) ∆ Φ( ) = Φ( m ) Φ , Φ , Φ(∆ )= (id A ⊗ m ⊗ id A )( α A, , ⊗ id A ) ◦ (id A ⊗ l ⊗ id A )(id A ⊗ ε A m A ∆ A u A ⊗ id A )(id A ⊗ l − ⊗ id A ) ◦ ( α − A, , ⊗ id A )(id A ⊗ ∆ ⊗ id A )= ϕ id Φ( ) , and ε Φ( ) u Φ( ) = Φ Φ( ε ) Φ( u ) Φ = m A ( r A ⊗ id A )(id A ⊗ ε u ⊗ id A )( r − A ⊗ id A )∆ A = id A . By the isomorphism χ above, and one can then rescale the multiplication of E toyield that E is special. (cid:3) By the proposition above, E ( A ) is a special Frobenius algebra in A C A , when A is special Frobenius. Now recall the functorΦ = Φ C A : C → A C A from Theorem 3.2, and consider the following functors: b Φ := Φ A C A E : A C A → E ( A C A ) E , b U : E ( A C A ) E → A C A (forget) ,U : A C A → C (forget) . Corollary 4.6.
The functors Φ , b Φ , U , b U are each monoidal and Morita preserving.Proof. We have that Φ is monoidal by Theorem 3.2, and b Φ is also monoidal byapplying Theorem 3.2 with Proposition 4.5. Moreover, it is straight-forward tocheck that U is monoidal with the following structure: for Y, Y ′ ∈ A C A , take(4.7) U Y,Y ′ = π AY,Y ′ : U ( Y ) ⊗ U ( Y ′ ) = Y ⊗ Y ′ → Y ⊗ A Y ′ = U ( Y ⊗ A Y ′ ) ,U = u A : → A = U ( A C A ) . For instance, the following diagram commutes due to the unit constraint on Y (asa left A -module in C ) and by definition of l AY : ⊗ U ( Y ) = ⊗ Y l Y (cid:15) (cid:15) U ⊗ id Y = u A ⊗ id Y / / A ⊗ Y = U ( A C A ) ⊗ U ( Y ) U A,Y = π AA,Y (cid:15) (cid:15) λ Y r r ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ U ( Y ) = Y A ⊗ A Y = U ( A ⊗ A Y ) l AY o o In a similar manner, the functor b U has a monoidal structure. LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 23
Next, we apply Theorem 4.1 to get that each of Φ, b Φ, U , b U are Morita preserv-ing. Indeed, it is clear from Theorem 3.2 that the natural transformations Φ ∗ , ∗ and b Φ ∗ , ∗ are epimorphisms. Moreover, Φ and b Φ are left adjoints (to U and b U ,respectively), so they preserve epimorphisms. On the other hand, we see that thenatural transformations U ∗ , ∗ and b U ∗ , ∗ are epimorphisms from (4.7). Lastly, U and b U preserve epimorphisms as they are faithful functors. (cid:3) Now we establish the main result of the section. But first we need a preliminaryresult on the algebra U b U b Φ Φ( B ) in C (resulting from the corollary above). Lemma 4.8.
For A a special Frobenius algebra in C and B ∈ Alg ( C ) , we get thefollowing statements. (a) D := U b U b Φ Φ( B ) ∈ Alg ( C ) . Here, D = ( E ⊗ A (( A ⊗ B ) ⊗ A )) ⊗ A E as anobject in C . (b) D is isomorphic to T := ( U ( E ) ⊗ B ) ⊗ U ( E ) = ( E ⊗ B ) ⊗ E as objects in C via θ := ( ν AE ⊗ id B ⊗ µ AE )( α − E,A,B ⊗ id AE ) α EAB,A,E ( α − E,AB,A ⊗ A id E ) : D ∼ → T, for natural isomorphisms µ AE : A ⊗ A E ∼ → E and ν AE : E ⊗ A A ∼ → E , andassociativity constraint α given in Lemma 2.24. (c) T admits the structure of an algebra in C , with m T = θ m D ( θ − ⊗ θ − ) and u T = θ u D , and D ∼ = T as algebras in C .Proof. Part (a) follows from Corollary 4.6 and Proposition 2.9. Part (b) holdsbecause µ AE and ν AE are isomorphisms in C (see [8, Exercise 7.8.22]), and α is anisomorphism in C by Lemma 2.24. Part (c) follows from parts (a) and (b). (cid:3) Theorem 4.9.
Take A a special Frobenius algebra in C , and take B, B ′ ∈ Alg ( C ) .Then, B and B ′ are Morita equivalent as algebras in C if and only if Φ( B ) and Φ( B ′ ) are Morita equivalent as algebras in A C A .Proof. The forward direction holds because Φ is Morita preserving by Corollary 4.6.For the converse, note that the algebras U b U b Φ Φ( B ) and U b U b Φ Φ( B ′ ) areMorita equivalent algebras in C because U , b U , b Φ are each Morita preserving [Corol-lary 4.6]. So it suffices to show that U b U b Φ Φ( B ) is Morita equivalent to B asalgebras in C , which we achieve as follows.By Lemma 4.8, D := U b U b Φ Φ( B ) is isomorphic to T = ( U ( E ) ⊗ B ) ⊗ U ( E ) =( E ⊗ B ) ⊗ E as algebras in C . So it suffices to show that T is Morita equivalentto B in C . This holds using the methods in the proof of Proposition 2.26(b). Wediscuss this in the strict case and leave the general case to the reader.We have by Remark 2.8 and Proposition 4.5 that E is a self-dual object in A C A with evaluation map p E = ε E m E : E ⊗ A E → A . To proceed, recall Notation 3.1and define the morphism φ := ε A e p E : E ⊗ E −→ . Now, take P = E ⊗ B with morphisms λ TP = (id E ⊗ m B )(id EB ⊗ φ ⊗ id B ) and ρ BP = (id E ⊗ m B ), and take Q = B ⊗ E with morphisms λ BQ = ( m B ⊗ id E ) and ρ TQ = ( m B ⊗ id E )(id B ⊗ φ ⊗ id BE ). We then obtain that P ∈ T C B and Q ∈ B C T .Moreover, we have epimorphisms b τ = id E ⊗ m B ⊗ id E : P ⊗ Q → T ∈ T C T , b µ = m B (id B ⊗ φ ⊗ id B ) : Q ⊗ P → B ∈ B C B , which factor through epimorphisms τ : P ⊗ B Q → T and µ : Q ⊗ T P → B ,respectively. Similar to the proof of Proposition 2.26(b), it is also straight-forwardto check that τ and µ satisfy diagrams ( ∗ ) and ( ∗∗ ) of Proposition 2.25(b). Thus,by Proposition 2.25, T and B are Morita equivalent in C , as desired. (cid:3) Algebras A ( L, ψ ) in pointed fusion categories Vec ωG We recall here the definition of a twisted group algebra in the pointed fusioncategory
Vec ωG [Definition 5.5]. We show that these algebras can be given thestructure of a Frobenius algebra in Vec ωG [Proposition 5.7], and further, that theyenjoy nice properties [Proposition 5.9]. Let us begin with discussing pointed tensorcategories. Definition 5.1.
Let C be a tensor category. It is called pointed if all of its simpleobjects are invertible, in the sense that the co/evaluation maps on simple objectsare isomorphisms in C .The following pointed fusion category will be crucial to our work. Definition 5.2 ( Vec ωG , δ g ) . Take G a finite group with 3-cocycle ω ∈ H ( G, k × ).The category Vec ωG is the category of G -graded vector spaces V = L x ∈ G V x withassociativity constraint ω given as follows. In particular, its simple objects are { δ g } g ∈ G , where the G -grading is ( δ g ) x = δ g,x · k , for g, x ∈ G . Morphisms are k -linear maps that preserve the G -grading.The monoidal structure is determined by the G -grading of objects( V ⊗ W ) x = M yz = x V y ⊗ W z , the associativity constraint α δ g ,δ g ′ ,δ g ′′ = ω − ( g, g ′ , g ′′ ) id δ gg ′ g ′′ : ( δ g ⊗ δ g ′ ) ⊗ δ g ′′ → δ g ⊗ ( δ g ′ ⊗ δ g ′′ ) , the unit object δ e , with unit constraints l δ g = ω − ( e, e, g ) id δ g , r δ g = ω ( g, e, e ) id δ g .The duals of simple objects are defined as δ ∗ g = δ g − = ∗ δ g , with evaluation mor-phisms given by ev δ g ( δ ∗ g ⊗ δ g ) = ω ( g, g − , g ) δ e and ev ′ δ g ( δ g ⊗ ∗ δ g ) = ω − ( g, g − , g ) δ e ,and coevaluation morphisms given by coev δ g ( δ e ) = δ g ⊗ δ ∗ g and coev ′ δ g ( δ e ) = ∗ δ g ⊗ δ g . Remark 5.3. (a) We assume as in [8, Remark 2.6.3] that ω is normalized; inparticular, l δ g = id δ g = r δ g . LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 25 (b) The associativity constraint α δ g ,δ g ′ ,δ g ′′ of Vec ωG given in [8] is defined by ω ( g, g ′ , g ′′ )id δ gg ′ g ′′ , but we need to use ω − ( g, g ′ , g ′′ ) id δ gg ′ g ′′ here in orderto get that the twisted group algebra A ( L, ψ ) presented in Definition 5.5below is an associative algebra in
Vec ωG .Not only is Vec ωG a pointed fusion category, we have that every pointed fusioncategory is equivalent to one of this type (see [10, Section 8.8]).For reference in computations later, the 3-cocycle condition on ω is(5.4) ω ( g g , g , g ) ω ( g , g , g g ) = ω ( g , g , g ) ω ( g , g g , g ) ω ( g , g , g )for all g i ∈ G .Next we turn our attention to algebras in, and module categories over, Vec ωG . Tocontinue, consider the following terminology. Definition 5.5 ( A ( L, ψ )) . Take L a subgroup of G so that ω | L × is trivial, andtake ψ ∈ C ( L, k × ) so that dψ = ω | L × . We assume that ψ is normalized. Wedefine the twisted group algebra A ( L, ψ ) in
Vec ωG to be L g ∈ L δ g as an object in Vec ωG ,with multiplication given by δ g ⊗ δ g ′ ψ ( g, g ′ ) δ gg ′ . It is well-known, and we will see later in Proposition 5.7, that A ( L, ψ ) is indeedan associative algebra in
Vec ωG .For reference in computations later, note that for a 2-cocycle, say θ , on a sub-group N of G the condition that dθ = ω | N × is translated as follows:(5.6) θ ( f , f f ) θ ( f , f ) = ω ( f , f , f ) θ ( f f , f ) θ ( f , f )for f i ∈ N .We show now that twisted group algebras A ( L, ψ ) are Frobenius algebras in
Vec ωG . Proposition 5.7.
The twisted group algebra A ( L, ψ ) admits the structure of aFrobenius algebra in Vec ωG : for g, g ′ ∈ L , it is given by m A ( L,ψ ) ( δ g ⊗ δ g ′ ) = ψ ( g, g ′ ) δ gg ′ ,u A ( L,ψ ) ( δ e ) = δ e , ∆ A ( L,ψ ) ( δ g ) = | L | − L h ∈ L ψ − ( gh, h − ) [ δ gh ⊗ δ h − ] ,ε A ( L,ψ ) ( δ g ) = δ g,e | L | δ e . Proof.
We start by showing that A ( L, ψ ) is an algebra in
Vec ωG . For the associativityof multiplication, consider the following calculation: m A ( L,ψ ) (id ⊗ m A ( L,ψ ) ) α [( δ g ⊗ δ g ′ ) ⊗ δ g ′′ ]= ω − ( g, g ′ , g ′′ ) m A ( L,ψ ) (id ⊗ m A ( L,ψ ) )[ δ g ⊗ ( δ g ′ ⊗ δ g ′′ )]= ω − ( g, g ′ , g ′′ ) ψ ( g ′ , g ′′ ) m A ( L,ψ ) ( δ g ⊗ δ g ′ g ′′ )= ω − ( g, g ′ , g ′′ ) ψ ( g ′ , g ′′ ) ψ ( g, g ′ g ′′ ) δ gg ′ g ′′ = ψ ( g, g ′ ) ψ ( gg ′ , g ′′ ) δ gg ′ g ′′ = ψ ( g, g ′ ) m A ( L,ψ ) ( δ gg ′ ⊗ δ g ′′ )= m A ( L,ψ ) ( m A ( L,ψ ) ⊗ id)[( δ g ⊗ δ g ′ ) ⊗ δ g ′′ ] . For the fourth equation, we used (5.6) with θ = ψ and ( f , f , f ) = ( g, g ′ , g ′′ ).Next, for u A ( L,ψ ) to satisfy the unit constraint, recall that ψ is normalized, and weget m A ( L,ψ ) ( u A ( L,ψ ) ⊗ id)( δ e ⊗ δ g ) = m A ( L,ψ ) ( δ e ⊗ δ g ) = ψ ( e, g ) δ g = δ g ,m A ( L,ψ ) (id ⊗ u A ( L,ψ ) )( δ g ⊗ δ e ) = m A ( L,ψ ) ( δ g ⊗ δ e ) = ψ ( g, e ) δ g = δ g . Thus, A ( L, ψ ) is an algebra in
Vec ωG .Next we define a nondegenerate pairing p , with copairing q , on A ( L, ψ ). Take p ( δ g ⊗ δ g ′ ) := ( | L | ψ ( g, g ′ ) δ e , gg ′ = e , gg ′ = e and q ( δ e ) := | L | − L h ∈ L ψ − ( h, h − ) [ δ h ⊗ δ h − ] . (5.8)Note that p (id ⊗ m ) α [( δ g ⊗ δ g ′ ) ⊗ δ g ′′ ] = ω − ( g, g ′ , g ′′ ) p (id ⊗ m )[ δ g ⊗ ( δ g ′ ⊗ δ g ′′ )]= ω − ( g, g ′ , g ′′ ) ψ ( g ′ , g ′′ ) p [ δ g ⊗ δ g ′ g ′′ ]= | L | ω − ( g, g ′ , g ′′ ) ψ ( g ′ , g ′′ ) ψ ( g, g ′ g ′′ ) δ gg ′ g ′′ ,e δ e = | L | ψ ( g, g ′ ) ψ ( gg ′ , g ′′ ) δ gg ′ g ′′ ,e δ e = ψ ( g, g ′ ) p [ δ gg ′ ⊗ δ g ′′ ]= p ( m ⊗ id) [( δ g ⊗ δ g ′ ) ⊗ δ g ′′ ] , where the fourth equality holds by (5.6) with θ = ψ and ( f , f , f ) = ( g, g ′ , g ′′ ).So, it suffices to verify the Snake Equation for p and q : r A (id A ⊗ p ) α A,A,A ( q ⊗ id A ) l − A = id A = l A ( p ⊗ id A ) α − A,A,A (id A ⊗ q ) r − A . Now,[ r A (id A ⊗ p ) α A,A,A ( q ⊗ id A ) l − A ]( δ g )= | L | − L h ∈ L ψ − ( h, h − ) r A (id A ⊗ p ) α A,A,A [( δ h ⊗ δ h − ) ⊗ δ g ]= | L | − L h ∈ L ψ − ( h, h − ) ω − ( h, h − , g ) r A (id A ⊗ p )[ δ h ⊗ ( δ h − ⊗ δ g )]= ψ − ( g, g − ) ω − ( g, g − , g ) ψ ( g − , g ) δ g = δ g , where the last equation holds by (5.6) for ( f , f , f ) = ( g, g − , g ). On the otherhand, [ l A ( p ⊗ id A ) α − A,A,A (id A ⊗ q ) r − A ]( δ g )= | L | − L h ∈ L ψ − ( h, h − ) l A ( p ⊗ id A ) α − A,A,A [ δ g ⊗ ( δ h ⊗ δ h − )]= | L | − L h ∈ L ψ − ( h, h − ) ω ( g, h, h − ) l A ( p ⊗ id A )[( δ g ⊗ δ h ) ⊗ δ h − )] LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 27 = ψ − ( g − , g ) ω ( g, g − , g ) ψ ( g, g − ) δ g = δ g , where, again, the last equation holds by (5.6) for ( f , f , f ) = ( g, g − , g ). Hence,by Remark 2.8, A ( L, ψ ) is a Frobenius algebra in
Vec ωG .For the Frobenius algebra ( A, m, u, p, q ) above, the comultiplication map ∆ andcounit map ε are given as follows (again due to Remark 2.8):∆( δ g ) = ( m ⊗ id A ) α − A,A,A (id A ⊗ q ) r − A ( δ g )= | L | − L h ∈ L ψ − ( h, h − ) ( m ⊗ id A ) α − [ δ g ⊗ ( δ h ⊗ δ h − )]= | L | − L h ∈ L ψ − ( h, h − ) ω ( g, h, h − ) ( m ⊗ id)[( δ g ⊗ δ h ) ⊗ δ h − ]= | L | − L h ∈ L ψ − ( h, h − ) ω ( g, h, h − ) ψ ( g, h ) [ δ gh ⊗ δ h − ]= | L | − L h ∈ L ψ − ( gh, h − ) [ δ gh ⊗ δ h − ] . Here, the ultimate equality holds by applying (5.6) to ( f , f , f ) = ( g, h, h − ).Moreover, ε ( δ g ) = p ( u ⊗ id A ) r − A ( δ g ) = p ( δ g ⊗ δ e ) = δ g,e | L | δ e . Therefore, A ( L, ψ ) ∈ FrobAlg ( Vec ωG ). (cid:3) Now we discuss algebraic properties of twisted group algebras; see Section 2.2.
Proposition 5.9.
The twisted group algebra A ( L, ψ ) , with structural morphisms m, u, ∆ , ε given in Proposition 5.7, possesses the following properties: (a) indecomposable; (b) connected; (c) separable; (d) special; and (e) symmetric Frobenius if and only if ω ( g − , g, g − ) = 1 for each g ∈ L .Proof. By Remark 2.11(b,d), parts (a) and (c) follow from parts (b) and (d), re-spectively. So we establish parts (b), (d), and (e) as follows. Denote A := A ( L, ψ ).(b) We have that Hom
Vec ωG ( Vec ωG , A ) = Hom Vec ωG ( δ e , ⊕ g ∈ L δ g ) = { δ e δ e } ,because morphisms preserve G -grading. Then dim Hom Vec ωG ( Vec ωG , A ) = 1 and A is connected.(d) The algebra A is special because m ∆( δ g ) = | L | − L h ∈ L ψ − ( gh, h − ) m [( δ gh ⊗ δ h − )]= | L | − L h ∈ L ψ − ( gh, h − ) ψ ( gh, h − ) δ g = δ g , and ε A ( L,ψ ) u A ( L,ψ ) ( δ e ) = | L | δ e = | L | id ( δ e ).(e) Since ∗ δ g = δ g − = δ ∗ g for all g ∈ L , we have that ∗ A = A ∗ as objects in Vec ωG .Moreover, recall Definition 2.12 and observe that εm ( δ g ⊗ δ g ′ ) = δ gg ′ ,e | L | ψ ( g, g ′ ) δ e , and that Ω A ( δ g ⊗ δ g ′ ) , is equal to [ ev A (id A ∗ ⊗ l A )(id A ∗ ⊗ εm ⊗ id A )( α ⊗ id A )(( coev ′ A ⊗ id A ) l − A ⊗ id A )]( δ g ⊗ δ g ′ )= L h ∈ L [ ev A (id A ∗ ⊗ l A )(id A ∗ ⊗ εm ⊗ id A )( α ⊗ id A )]((( δ h − ⊗ δ h ) ⊗ δ g ) ⊗ δ g ′ )= L h ∈ L ω − ( h − , h, g )[ ev A (id A ∗ ⊗ l A )(id A ∗ ⊗ εm ⊗ id A )](( δ h − ⊗ ( δ h ⊗ δ g )) ⊗ δ g ′ )= | L | L h ∈ L δ hg,e ω − ( h − , h, g ) ψ ( h, g ) ev A ( δ h − ⊗ δ g ′ )= | L | ω − ( g, g − , g ) ψ ( g − , g ) ev A ( δ g ⊗ δ g ′ )= ⊕ t ∈ L | L | ω − ( g, g − , g ) ψ ( g − , g ) ev δ t ( δ g ⊗ δ g ′ )= δ gg ′ ,e | L | ω − ( g, g − , g ) ψ ( g − , g ) ev δ g − ( δ ∗ g − ⊗ δ g − )= δ gg ′ ,e | L | ω − ( g, g − , g ) ψ ( g − , g ) ω ( g − , g, g − ) δ e . Since A is Frobenius by Proposition 5.7, by the computations above we get that A is symmetric Frobenius if and only if if and only if ψ ( g, g − ) = ω − ( g, g − , g ) ψ ( g − , g ) ω ( g − , g, g − ) . From (5.6) with f = g, f = g − , f = g , this is equivalent to ω ( g − , g, g − ) = 1for all g ∈ L . (cid:3) Algebras A K,β ( L, ψ ) in group-theoretical fusion categories C ( G, ω, K, β )We define in this section the main structures of interest in this work: twistedHecke algebras [Definition 6.3]. These are algebras in group-theoretical fusion cat-egories C [Definition 6.1] that are analogous to the twisted group algebras in Vec ωG discussed in Section 5. We establish that the twisted Hecke algebras admit thestructure of a Frobenius algebra in C [Theorem 6.4], and further, as algebras in C weshow that they are indecomposable, separable, and special [Proposition 6.10]. Wealso discuss when these (Frobenius) algebras are connected in C [Proposition 6.13,Remark 6.19].We begin by introducing the terminology mentioned above. Definition 6.1 ( C ( G, ω, K, β )) . [10, Section 8.8; Definition 8.40] A group-theoreticalfusion category is a category of bimodules of the form C ( G, ω, K, β ) := A ( K,β ) ( Vec ωG ) A ( K,β ) for a twisted group algebra A ( K, β ) in
Vec ωG .This is equivalent to the functor category Fun
Vec ωG ( M ( K, β ) , M ( K, β )) op ; see[8, Proposition 7.11.1, Definition 7.12.2, and Remark 7.12.5]. Next, we recall adescription of simple objects of group-theoretical fusion categories. LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 29
Lemma 6.2. [8, Example 9.7.4] [20, Section 5]
Any simple object of C ( G, ω, K, β ) is of the form V g,ρ = M f ∈ K,k ∈ T ( δ f ⊗ δ g ) ⊗ δ k ⊕ n g , where g ∈ G is a representative of a double coset in K \ G/K , T is a set of repre-sentatives of the classes in K/K g − for K g − := ( K ∩ g − Kg ) , ρ : K g − → GL( V ) is a certain irreducible projective representation, and n g = dim V . The A ( K, β ) -bimodule structure on V g,ρ is given by the left A ( K, β ) -action m A ( K,β ) ⊗ id ⊗ id and the compatible right A ( K, β ) -action is determined by the left A ( K, β ) -actionand ρ . (cid:3) Now we turn our attention to algebraic structures in group-theoretical fusioncategories.
Definition 6.3 ( A K,β ( L, ψ )) . Consider the functorΦ :
Vec ωG → C ( G, ω, K, β )from Theorem 3.2 in the case when C = Vec ωG and A = A ( K, β ). We refer toΦ( A ( L, ψ )) =: A K,β ( L, ψ )as a twisted Hecke algebra in C ( G, ω, K, β ).We use this terminology because the simple objects of the group-theoreticalfusion category C ( G, ω, K, β ) are in part parameterized by K -double cosets in G [Lemma 6.2], and as we see below, the multiplication of the algebra is twisted bycocycles. Theorem 6.4.
The twisted Hecke algebra A K,β ( L, ψ ) equals M g ∈ L ; f,k ∈ K ( δ f ⊗ δ g ) ⊗ δ k as an object in C ( G, ω, K, β ) . Furthermore, for f, f ′ , k, k ′ , d, d ′ ∈ K and g, g ′ ∈ L ,we have the following statements. (a) A K,β ( L, ψ ) has the structure of an algebra in C ( G, ω, K, β ) , where m A K,β ( L,ψ ) [(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ A ( K,β ) (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]= δ kf ′ ,e ω ( f gk, f ′ g ′ , k ′ ) ω − ( f g, k, f ′ g ′ ) ω ( k, f ′ , g ′ ) ω − ( f, g, g ′ ) · β ( k, f ′ ) ψ ( g, g ′ ) [( δ f ⊗ δ gg ′ ) ⊗ δ k ′ ] ,u A K,β ( L,ψ ) ( δ d ) = L s ∈ K β − ( ds − , s ) [( δ ds − ⊗ δ e ) ⊗ δ s ] . (b) With the above, A K,β ( L, ψ ) is a Frobenius algebra in C ( G, ω, K, β ) , where ∆ A K,β ( L,ψ ) [( δ f ⊗ δ g ) ⊗ δ k ]= | K | − | L | − L h ∈ L ; s ∈ K ω ( f, gh, h − ) ω ( f gh, s, s − h − ) ω − ( f ghs, s − h − , k ) · ω − ( s, s − , h − ) ψ − ( gh, h − ) β − ( s, s − ) · [(( δ f ⊗ δ gh ) ⊗ δ s ) ⊗ A ( K,β ) (( δ s − ⊗ δ h − ) ⊗ δ k )] ,ε A K,β ( L,ψ ) [( δ f ⊗ δ g ) ⊗ δ k ] = δ g,e | L | β ( f, k ) δ fk . Proof.
The definition of the functor Φ gives us A K,β ( L, ψ ) := Φ( A ( L, ψ )) = ( A ( K, β ) ⊗ A ( L, ψ )) ⊗ A ( K, β ) , which corresponds to the object M g ∈ L ; f,k ∈ K ( δ f ⊗ δ g ) ⊗ δ k in the category C ( G, ω, K, β ). Throughout this proof, we will fix the notation A := A ( K, β ) and B := A ( L, ψ )for simplicity. Recall that Φ( f ) = id A ⊗ f ⊗ id A for any morphism f in Vec ωG .(a) Since the functor Φ is monoidal [Theorem 3.2], we have by Proposition 2.9(a)that A K,β ( L, ψ ) = Φ( B ) is an algebra in C ( G, ω, K, β ), with multiplication andunit maps given by Φ( m B )Φ B,B and Φ( u B )Φ , respectively. Here, the monoidalstructure of Φ is defined in Theorem 3.2, and in particular, the morphism Φ B,B isgiven by means of the lift e Φ B,B [Notation 3.1].Note that e Φ B,B [(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ ( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]= ( α A,B,B ⊗ id A ) (id A,B ⊗ l B ⊗ id A ) (id A,B ⊗ ε A m A ⊗ id B,A )(id
A,B ⊗ α − A,A,B ⊗ id A ) ( α AB,A,AB ⊗ id A ) α − ABA,AB,A [(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]= ω ( f gk, f ′ g ′ , k ′ ) ω − ( f g, k, f ′ g ′ ) ω ( k, f ′ , g ′ )( α A,B,B ⊗ id A ) (id A,B ⊗ l B ⊗ id A ) (id A,B ⊗ ε A m A ⊗ id B,A )[(( δ f ⊗ δ g ) ⊗ ( δ k ⊗ ( δ f ′ ⊗ δ g ′ ))) ⊗ δ k ′ ]= δ kf ′ ,e ω ( f gk, f ′ g ′ , k ′ ) ω − ( f g, k, f ′ g ′ ) ω ( k, f ′ , g ′ ) β ( k, f ′ )( α A,B,B ⊗ id A )(id A,B ⊗ l B ⊗ id A )[(( δ f ⊗ δ g ) ⊗ ( δ e ⊗ δ g ′ )) ⊗ δ k ′ ]= δ kf ′ ,e ω ( f gk, f ′ g ′ , k ′ ) ω − ( f g, k, f ′ g ′ ) ω ( k, f ′ , g ′ ) β ( k, f ′ ) ω − ( f, g, g ′ )( δ f ⊗ ( δ g ⊗ δ g ′ )) ⊗ δ k ′ . Therefore, the multiplication of A K,β ( L, ψ ) is given by m A K,β ( L,ψ ) [(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )] LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 31 = δ kf ′ ,e ω ( f gk, f ′ g ′ , k ′ ) ω − ( f g, k, f ′ g ′ ) ω ( k, f ′ , g ′ ) ω − ( f, g, g ′ ) β ( k, f ′ ) ψ ( g, g ′ ) [( δ f ⊗ δ gg ′ ) ⊗ δ k ′ ] . Here, we use Proposition 5.7 for the multiplication and counit of A , and themonoidal structure of Vec ωG is given in Definition 5.2.On the other hand, by using the definition of Φ from Theorem 3.2 we get thatthe unit of A K,β ( L, ψ ) is given by u A K,β ( L,ψ ) ( δ d ) = Φ( u B ) Φ ( δ d ) = Φ( u B ) ( r − A ⊗ id A ) ∆ A ( δ d )= Φ( u B ) ( r − A ⊗ id A ) L s ∈ K β − ( s − , s ) β ( d, s ) ω ( s, s − , s ) ω ( d, s, s − ) [ δ ds ⊗ δ s − ]= L s ∈ K β − ( s − , s ) β ( d, s ) ω ( s, s − , s ) ω ( d, s, s − ) [( δ ds ⊗ δ e ) ⊗ δ s − ]= L s ∈ K β − ( s, s − ) β ( d, s − ) ω ( s − , s, s − ) ω ( d, s − , s ) [( δ ds − ⊗ δ e ) ⊗ δ s ]= L s ∈ K β − ( s, s − ) β ( d, s − ) ω ( ds, s, s − ) [( δ ds − ⊗ δ e ) ⊗ δ s ]= L s ∈ K β − ( ds − , s ) [( δ ds − ⊗ δ e ) ⊗ δ s ] . For the penultimate equation we used (5.4) with ( g , g , g , g ) = ( d, s − , s, s − ),and we used (5.6) with θ = ψ and ( f , f , f ) = ( ds − , s, s − ) for the last equation.Moreover, we use Proposition 5.7 for the comultiplication of A , and again themonoidal structure of Vec ωG is described in Definition 5.2.(b) Since the functor Φ is Frobenius monoidal (see Theorem 3.2) and B is aFrobenius algebra in Vec ωG (see Proposition 5.7), A K,β ( L, ψ ) = Φ( B ) is a Frobeniusalgebra in C ( G, ω, K, β ) by Proposition 2.9(c). Moreover, the comultiplication andcounit of A K,β ( L, ψ ) determined by Φ are Φ
B,B
Φ(∆ B ) and Φ Φ( ε B ), respectively.Recall that the comonoidal structure Φ ∗ , ∗ and Φ of Φ is described in Theorem 3.2,and the structure of A and B are given in Proposition 5.7. Now,(6.5) Φ B,B
Φ(∆ B ) [( δ f ⊗ δ g ) ⊗ δ k ]= | L | − L h ∈ L ψ − ( gh, h − ) Φ B,B [( δ f ⊗ ( δ gh ⊗ δ h − )) ⊗ δ k ] . Moreover, the lift e Φ B,B of Φ
B,B on ( δ f ⊗ ( δ gh ⊗ δ h − )) ⊗ δ k is given as follows: e Φ B,B [( δ f ⊗ ( δ gh ⊗ δ h − )) ⊗ δ k ]= α ABA,AB,A ( α − AB,A,AB ⊗ id A ) (id A,B ⊗ α A,A,B ⊗ id A ) (id A,B ⊗ ∆ A ⊗ id B,A )(id
A,B ⊗ u A l − B ⊗ id A ) ( α − A,B,B ⊗ id A ) [( δ f ⊗ ( δ gh ⊗ δ h − )) ⊗ δ k ]= α ABA,AB,A ( α − AB,A,AB ⊗ id A ) (id A,B ⊗ α A,A,B ⊗ id A )(id A,B ⊗ ∆ A ⊗ id B,A ) ω ( f, gh, h − )[(( δ f ⊗ δ gh ) ⊗ ( δ e ⊗ δ h − )) ⊗ δ k ]= L s ∈ K ω ( f, gh, h − ) ω ( s, s − , s ) β − ( s − , s ) α ABA,AB,A ( α − AB,A,AB ⊗ id A )(id A,B ⊗ α A,A,B ⊗ id A )[(( δ f ⊗ δ gh ) ⊗ (( δ s ⊗ δ s − ) ⊗ δ h − )) ⊗ δ k ]= L s ∈ K ω ( f, gh, h − ) ω ( s, s − , s ) ω − ( s, s − , h − ) ω ( f gh, s, s − h − ) ω − ( f ghs, s − h − , k ) β − ( s − , s ) [(( δ f ⊗ δ gh ) ⊗ δ s ) ⊗ (( δ s − ⊗ δ h − ) ⊗ δ k )] , = L s ∈ K ω ( f, gh, h − ) ω ( f gh, s, s − h − ) ω − ( f ghs, s − h − , k ) ω − ( s, s − , h − ) β − ( s, s − ) [(( δ f ⊗ δ gh ) ⊗ δ s ) ⊗ (( δ s − ⊗ δ h − ) ⊗ δ k )];here, we used (5.6) with θ = β and ( f , f , f ) = ( s, s − , s ) for the last equation.Together, with (6.5), we can normalize Φ B,B Φ B (∆ B ) by multiplying by | K | − toget the desired formula for ∆ A K,β ( L,ψ ) .On the other hand, ε A K,β ( L,ψ ) [( δ f ⊗ δ g ) ⊗ δ k ]= Φ Φ( ε B )[( δ f ⊗ δ g ) ⊗ δ k ]= δ g,e | L | m A ( r A ⊗ id A )[( δ f ⊗ δ e ) ⊗ δ k ]= δ g,e | L | β ( f, k ) δ fk . (cid:3) We include a direct proof of the theorem above in Appendix C.
Remark 6.6.
Taking the forgetful functor U : C ( G, ω, h e i , → Vec ωG , observe that U ( A h e i , ( L, ψ )) ∼ = A ( L, ψ ) as algebras in
Vec ωG .The next result is expected, but we include it for the interest of the reader. Proposition 6.7. If ψ and ψ ′ are cohomologous -cocycles in Z ( L, k × ) , then weget that A K,β ( L, ψ ) ∼ = A K,β ( L, ψ ′ ) as algebras in C ( G, ω, K, β ) .Proof. Let γ : L → k × be a 1-cochain such that dγ = ψ/ψ ′ , that is,(6.8) γ ( g ) γ ( g ) γ − ( g g ) = ψ ( g , g ) ψ ′− ( g , g ) for all g , g ∈ G. Let A := A ( K, β ) , A := A K,β ( L, ψ ) , A := A K,β ( L, ψ ′ ) and let φ : A → A bedefined by φ (( δ f ⊗ δ g ) ⊗ δ k ) = γ ( g ) ( δ f ⊗ δ g ) ⊗ δ k , which is clearly an A -bimodule map. We get that m A ( φ ⊗ A φ )[(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ ))]= m A ( γ ( g ) γ ( g ′ ) [( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )) ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ ))])= γ ( g ) γ ( g ′ ) δ kf ′ ,e ω ( f gk, f ′ g ′ , k ′ ) ω − ( f g, k, f ′ g ′ ) ω ( k, f ′ , g ′ ) ω − ( f, g, g ′ ) · β ( k, f ′ ) ψ ′ ( g, g ′ ) [( δ f ⊗ δ gg ′ ) ⊗ δ k ′ ] . On the other hand, φ ( m A )[(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]= φ ( δ kf ′ ,e ω ( f gk, f ′ g ′ , k ′ ) ω − ( f g, k, f ′ g ′ ) ω ( k, f ′ , g ′ ) ω − ( f, g, g ′ ) · β ( k, f ′ ) ψ ( g, g ′ ) [( δ f ⊗ δ gg ′ ) ⊗ δ k ′ ])= γ ( gg ′ ) δ kf ′ ,e ω ( f gk, f ′ g ′ , k ′ ) ω − ( f g, k, f ′ g ′ ) ω ( k, f ′ , g ′ ) ω − ( f, g, g ′ ) · β ( k, f ′ ) ψ ( g, g ′ ) [( δ f ⊗ δ gg ′ ) ⊗ δ k ′ ] . LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 33
These are equal due to (6.8). Concerning the unit map in A , we have that φ ( u A )( δ d ) = L s ∈ K β − ( ds − , s ) [( δ ds − ⊗ δ e ) ⊗ δ s ] = u A ( δ d ) . since γ ( e ) = 1. So, φ is an algebra morphism. The inverse morphism φ − : A → A is given by φ − ( δ f ⊗ δ g ⊗ δ k ) = γ ( g ) − δ f ⊗ δ g ⊗ δ k . Therefore, A ∼ = A as algebras in C ( G, ω, K, β ). (cid:3) Next, we discuss algebraic properties of twisted Hecke algebras; see Section 2.2.
Lemma 6.9.
Any subalgebra S of A K,β ( L, ψ ) contains, as objects in C ( G, ω, K β ) ,a direct summand of the form, S ( g ) := M f,k ∈ Kt =1 ,..., ord ( g ) ( δ f ⊗ δ g t ) ⊗ δ k , for every g ∈ L .Proof. Let A := A ( K, β ). Consider the two cases: S contains a direct summand ofthe form(1) ( δ c ⊗ δ g ) ⊗ δ c − for some c ∈ K and g ∈ L , or(2) ( δ f ⊗ δ g ) ⊗ δ k for some f, k ∈ K with f k = e and g ∈ L .Now in order for the multiplication of S to be induced by that of A K,β ( L, ψ ) incase (1), we must have that m ((( δ c ⊗ δ g ) ⊗ δ c − ) ⊗ A (( δ c ⊗ δ g ) ⊗ δ c − )) = λ (( δ c ⊗ δ g ) ⊗ δ c − )belongs to S , for some nonzero scalar λ . By repeating this argument, we have that S contains a direct summand of the form S ( g, c ) := M t ∈ N ( δ c ⊗ δ g t ) ⊗ δ c − . In order for the multiplication of S to be induced by that of A K,β ( L, ψ ) incase (2), we must have that m ((( δ f ⊗ δ g ) ⊗ δ k ) ⊗ A (( δ f ⊗ δ g ) ⊗ δ k )) = 0belongs to S . So, we just get that S contains ( δ f ⊗ δ g ) ⊗ δ k as a direct summand.On the other hand, in order for the unit of S to be induced by that of A K,β ( L, ψ )in case (1), we must have that m ( u ⊗ A id)( δ d ⊗ A (( δ c ⊗ δ g t ) ⊗ δ c − )) = λ d,g,t ( δ dc ⊗ δ g t ) ⊗ A δ c − belongs to S . By repeating this argument, and by using m (id ⊗ u ), we have that S contains a direct summand of the form S ( g ).Finally, for the unit of S to be induced by that of A K,β ( L, ψ ) in case (2), wemust have that m ( u ⊗ A id)( δ d ⊗ A (( δ f ⊗ δ g ) ⊗ δ k )) = λ d ( δ df ⊗ δ g ) ⊗ δ k belongs to S for some nonzero scalar λ d , for every d ∈ K . By taking d = k − f − ,we are in case (1) and S then contains a direct summand of the form S ( g ) by theargument above. (cid:3) Proposition 6.10.
The twisted Hecke algebra A K,β ( L, ψ ) , with structural mor-phisms m, u, ∆ , ε given in Theorem 6.4, possesses the following properties: (a) indecomposable; (b) separable; and (c) special.Proof. (a) This follows from Remark 2.11(a) and Lemma 6.9. Namely, the inter-section of any two subalgebras S ( g ) and S ( g ′ ) from Lemma 6.9 is never trivial.(b) This follows from Remark 2.11(d) and part (c) below.(c) To verify the special property, we compute: m A K,β ( L,ψ ) ∆ A K,β ( L,ψ ) [( δ f ⊗ δ g ) ⊗ δ k ]= | K | − | L | − L h ∈ L ; s ∈ K ω ( f, gh, h − ) ω ( f gh, s, s − h − ) ω − ( f ghs, s − h − , k ) · ω − ( s, s − , h − ) ψ − ( gh, h − ) β − ( s, s − ) · m A K,β ( L,ψ ) [(( δ f ⊗ δ gh ) ⊗ δ s ) ⊗ A ( K,β ) (( δ s − ⊗ δ h − ) ⊗ δ k )]= | K | − | L | − L h ∈ L ; s ∈ K ω ( f, gh, h − ) ω ( f gh, s, s − h − ) ω − ( f ghs, s − h − , k ) · ω − ( s, s − , h − ) ψ − ( gh, h − ) β − ( s, s − ) · ω ( f ghs, s − h − , k ) ω − ( f gh, s, s − h − ) ω ( s, s − , h − ) · ω − ( f, gh, h − ) ψ ( gh, h − ) β ( s, s − ) (( δ f ⊗ δ g ) ⊗ δ k )= | K | − | L | − L h ∈ L ; s ∈ K ( δ f ⊗ δ g ) ⊗ δ k = ( δ f ⊗ δ g ) ⊗ δ k , and for A K,β ( L, ψ ) = L d ∈ K δ d , we get ε A K,β ( L,ψ ) u A K,β ( L,ψ ) ( δ d )= ε A K,β ( L,ψ ) (cid:0)L s ∈ K β − ( ds − , s )[( δ ds − ⊗ δ e ) ⊗ δ s ] (cid:1) = L s ∈ K β − ( ds − , s ) | L | β ( ds − , s ) δ d = | K | | L | δ d . Therefore, m A K,β ( L,ψ ) ∆ A K,β ( L,ψ ) = id A K,β ( L,ψ ) , ε A K,β ( L,ψ ) u A K,β ( L,ψ ) = | K | | L | id A ( K,β ) . (cid:3) Now in comparison with Proposition 5.9(e), we ask:
Question 6.11.
Under what conditions are the twisted Hecke algebras symmetric?In order to address this question, see Definition 2.12 and consider Question 2.22.Now we examine the connected property of A K,β ( L, ψ ). LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 35
Definition 6.12 ( r ( β )) . Let β : K × K → k be a 2-cocycle, and suppose that K = { k , . . . , k m } . Let M β be the m × m - matrix with entries m ij = β ( k i , k j ). Wedefine r ( β ) to be the rank of M β . Proposition 6.13.
For the twisted Hecke algebra A K,β ( L, ψ ) , it holds that dim k (Hom A C A ( A, A
K,β ( L, ψ ))) = r ( β − ) . As a consequence, A K,β ( L, ψ ) is connected precisely when r ( β − ) = 1 .Proof. Take A := A ( K, β ) and B := A ( L, ψ ). Let φ : A → A K,β ( L, ψ ) be anonzero morphism in C ( G, ω, K, β ). Since A is a simple object in C ( G, ω, K, β ), it isindecomposable. So, image( φ ) is indecomposable. Since C ( G, ω, K, β ) is semisimple,image( φ )= V g,ρ from Lemma 6.2 for some pair ( g, ρ ) with g ∈ L . Since φ mustpreserve degree, φ ( δ d ) = M f ∈ K ; g ∈ L ; k ∈ K/ ( K ∩ g − Kg ); fgk = d λ f,g,k ( δ f ⊗ δ g ) ⊗ δ k . Then, g ∈ K , and the value above is equal to φ ( δ d ) = M f,g ∈ K ; k ∈ K/ ( K ∩ g − Kg ); fgk = d λ f,g,k ( δ fg ⊗ δ e ) ⊗ δ k = M s ∈ K λ d,s ( δ ds − ⊗ δ e ) ⊗ δ s . Therefore φ sends A to V e,ρ , and we have that φ : A → A K,β ( L, ψ ) , δ d L s ∈ K λ d,s (( δ ds − ⊗ δ e ) ⊗ δ s ) , for some scalars λ d,s not all zero.Next, using the fact that φ is a left A -module map, we get φ m A = ( m A ⊗ id B ⊗ id A ) α − A,A,B ⊗ id A ) α − A,AB,A (id A ⊗ φ ) . Applying both sides to δ d ′ ⊗ δ d yields λ d,s ω ( d ′ , ds − , s ) β ( d ′ , ds − ) = λ d ′ d,s β ( d ′ , d ) . By using (5.6) with θ = β and ( f , f , f ) = ( d ′ , ds − , s ), we get that(6.14) λ d,s β − ( d ′ ds − , s ) = λ d ′ d,s β − ( ds − , s ) . Likewise, φ being a right A -module yields that(6.15) λ d,s β − ( ds − , sd ′′ ) = λ dd ′′ ,s β − ( ds − , s ) . Solving for λ d ′ d,s in (6.14) and for λ dd ′′ ,s in (6.15), and taking d = e we get(6.16) λ d ′ ,s = λ e,s β ( s − , s ) β − ( d ′ s − , s ) , for all d ′ ∈ K (6.17) λ d ′′ ,s = λ e,s β ( s − , s ) β − ( s − , sd ′′ ) , for all d ′′ ∈ K. That is, in either case, all the constants λ d,s for d ∈ K depend on λ e,s (which cantake any value in k ) and the values of the 2-cocycle β . We proceed using (6.16);the argument using (6.17) yields the same conclusion in an analogous way.Let λ e,s = r s β − ( s − , s ), for r s ∈ k . Then, by (6.16), φ ( δ d ) = M s ∈ K λ d,s δ ds − ⊗ δ e ⊗ δ s = M s ∈ K λ e,s β ( s − , s ) β − ( ds − , s ) δ ds − ⊗ δ e ⊗ δ s = M s ∈ K r s β − ( ds − , s ) δ ds − ⊗ δ e ⊗ δ s . (6.18)Thus, the morphism φ can be identified with a diagonal matrix of size | K | with i -thentry equal to the scalar M s ∈ K r s β − ( d i s − , s ) . (Note that if r s = 1 for all s , then φ is the unit map of A K,β ( L, ψ ).) Define φ j ∈ Hom A C A ( A, A
K,β ( L, ψ )) by taking r s j = δ s,s j in (6.18), for each s j ∈ K . Thatis, φ j ( δ d i ) = β − ( d i s − j , s j ) ( δ d i s − j ⊗ δ e ⊗ δ s j ) , for all d i ∈ K, s j ∈ K. Then φ is a linear combination of the φ j ’s. More precisely, φ = M s j ∈ K r s j φ j . Let M β − be the matrix with entries m ij = β − ( d i s − j , s j ). Then, the number ofindependent morphisms among the φ j ’s is the rank of M β − , as desired. (cid:3) Remark 6.19.
Recall that in the special case when K = h e i , β = 1, the twistedHecke algebra A K,β ( L, ψ ) is, via Remark 6.6, the twisted group algebra A ( L, ψ ).Here, dim k Hom A C A ( A, A
K,β ( L, ψ )) = r ( β − ) = 1, so A ( L, ψ ) is connected. Thisrecovers Proposition 5.9(b).7.
Representation theory of group-theoretical fusion categories
We provide in this section a classification of indecomposable semisimple rep-resentations of group-theoretical fusion categories in terms of the twisted Heckealgebras defined and studied in Section 6; see Proposition 2.15 and Theorem 7.4below. This result is analogous to Ostrik and Natale’s classification of indecom-posable semisimple representations of pointed fusion categories in terms of twistedgroup algebras (studied in Section 5) [29, 28]; see Theorem 7.3 below.To begin, recall the notation from Sections 5 and 6, and consider the followingnotation.
Notation 7.1 ( x s , x S , ψ x , Ω x , M ( L, ψ ), M K,β ( L, ψ )) . . • We write x s := xsx − and x S := { x s : s ∈ S } , for x ∈ G and any set S . LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 37 • Take a 2-cochain ψ on a subgroup L of G and an element x ∈ G . The2-cochain ψ x on x L is defined by ψ x ( h , h ) = ψ ( x h , x h ) for h , h ∈ L . • For x ∈ G , define the 2-cocycle Ω x : G × G → k × byΩ x ( h , h ) = ω ( x h , x h , x ) ω ( x, h , h ) ω ( x h , x, h ) . • Let M ( L, ψ ) denote the left
Vec ωG -module category consisting of right A ( L, ψ )-modules in
Vec ωG . • Let M K,β ( L, ψ ) denote the left C ( G, ω, K, β )-module category consisting ofright A K,β ( L, ψ )-modules in C ( G, ω, K, β ).Next, we borrow a condition from [28].
Definition 7.2 ( P ( G, ω )) . Let L , L ′ be subgroups of G . Take ψ ∈ C ( L, k × ) with dψ = ω | L , and take ψ ′ ∈ C ( L ′ , k × ) with dψ ′ = ω | L ′ . We say that the pairs ( L, ψ )and ( L ′ , ψ ′ ) are conjugate if there exists an element x ∈ G so that(a) L = x L ′ , and(b) the class of the 2-cocycle ψ ′− ψ x Ω x | L ′ × L ′ is trivial in H ( L ′ , k × ).We denote by P ( G, ω ) the set of conjugacy classes of pairs (
L, ψ ) as above.Now consider the classification result for representations of pointed fusion cate-gories mentioned above.
Theorem 7.3. [29, Example 2.1] [8, Example 9.7.2] [28] . (a) We have that M ( L, ψ ) and M ( L ′ , ψ ′ ) are equivalent as Vec ωG -module cate-gories if and only if ( L, ψ ) = ( L ′ , ψ ′ ) in P ( G, ω ) . (b) Every indecomposable, semisimple left module category over the pointed fu-sion category
Vec ωG is equivalent to one of the form M ( L, ψ ) , as left Vec ωG -module categories. (cid:3) This brings us to the main result of this section, and of this article.
Theorem 7.4.
We have the following statements. (a) M K,β ( L, ψ ) and M K,β ( L ′ , ψ ′ ) are equivalent as C ( G, ω, K, β ) -module cate-gories if and only if ( L, ψ ) = ( L ′ , ψ ′ ) in P ( G, ω ) . (b) Every indecomposable, semisimple left module category over C ( G, ω, K, β ) is equivalent to one of the form M K,β ( L, ψ ) , as left C ( G, ω, K, β ) -modulecategories.Proof. (a) By Theorem 7.3, we need to show that M ( L, ψ ) and M ( L ′ , ψ ′ ) areequivalent as Vec ωG -module categories if and only if M K,β ( L, ψ ) and M K,β ( L ′ , ψ ′ )are equivalent as C ( G, ω, K, β )-module categories. But this holds by using The-orem 4.9, with Propositions 5.7 and 5.9(d), applied to C = Vec ωG , A = A ( K, β ), B = A ( L, ψ ), and B ′ = A ( L ′ , ψ ′ ).(b) For a fusion category D , let Indec ( Mod ( D )) denote a set of equivalence classrepresentatives of indecomposable semisimple left D -module categories, and let [ M ]be the class of D -module categories equivalent to M (as left D -module categories). Now by Theorem 7.3 and [26, Sections 3 and 4] (see also [8, Theorem 7.12.11]),there is a 1-to-1 correspondence between the finite sets,
Indec ( Mod ( C ( G, ω, K, β ))) and P ( G, ω );namely, both of these sets are in bijection with
Indec ( Mod ( Vec ωG )). On the otherhand, since A K,β ( L, ψ ) is an indecomposable and separable algebra in C ( G, ω, K, β )[Proposition 6.10], the finite collection { [ M K,β ( L, ψ )] } ( L,ψ ) ∈P ( G,ω ) consists of equivalence classes of indecomposable semisimple left C ( G, ω, K, β )-module categories [Proposition 2.15]. (Indeed, indecomposability and semisim-plicity are preserved under module category equivalence.) Moreover, by (a), thiscollection is also in bijection with the finite set P ( G, ω ). Therefore, as finite sets,
Indec ( Mod ( C ( G, ω, K, β ))) = { [ M K,β ( L, ψ )] } ( L,ψ ) ∈P ( G,ω ) , and this verifies part (b). (cid:3) To compare the twisted Hecke algebras in C ( G, ω, K, β ) with the algebras inProposition 2.26, we have the following remark.
Remark 7.5. If A is an algebra in a rigid monoidal category C such that A = A ∗ = ∗ A and B is a coalgebra in C , then we can give a coalgebra structure in ( ∗ A ⊗ B ) ⊗ A such that this is Frobenius when B is Frobenius. Also, if A and B are Frobeniusin C with A special, then ( ∗ A ⊗ B ) ⊗ A is isomorphic to ( A ⊗ B ) ⊗ A in A C A andtherefore is Frobenius. Now by Propositions 2.26, 5.7, and 5.9(d), and Theorem 7.4,the (Frobenius) algebras ( ∗ A ( K, β ) ⊗ A ( L, ψ )) ⊗ A ( K, β ) serve as Morita equivalenceclass representatives of indecomposable, separable algebras in C ( G, ω, K, β ).Finally, we compare our work with recent work of P. Etingof, R. Kinser, and thelast author in [9].
Remark 7.6.
Morita equivalence class representatives of indecomposable, separa-ble algebras in group-theoretical fusion categories C were used in the recent studyof tensor algebras in C ; see [9, Theorem 3.11 and Section 5]. (Note that a ‘separablealgebra’ here is the same as a ‘semisimple algebra’ in [9] as we are working over analgebraically closed field.) Now by Theorem 7.4, our construction of the twistedHecke algebras in C serve as the base algebras of tensor algebras in C , up to thenotion of equivalence given in [9, Definition 3.4]. Example 7.7.
Continuing the remark above, let
Rep ( H ) be the category of finite-dimensional representations of the Kac-Paljutkin Hopf algebra, which is a group-theoretical fusion category C ( D , ω, Z , Rep ( H ) is given in [9, Theorem 5.23]. The correspondence of thosesix algebras with the conjugacy classes of pairs ( L, ψ ) is presented in [9, Proposi-tion 5.26]. Thus, we can replace the algebras in [9, Theorem 5.23] corresponding tosuch pairs (
L, ψ ) with the twisted Hecke algebras A Z , ( L, ψ ) featured here. The
LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 39 advantage is that the six algebras of [9, Theorem 5.23] were found via ad-hoc meth-ods [9, Remark 5.28], whereas our construction provides a uniform collection ofMorita equivalence classes representatives of algebras in
Rep ( H ). Appendix A. Remainder of the proof of Theorem 3.2
In this appendix, we fill in various details for the proof of Theorem 3.2: Propo-sitions A.1 and A.2 below are for the rest of condition (a); Proposition A.3 is forthe rest of condition (b.1); and Propositions A.4–A.9 are for conditions (c)–(h),respectively. To make the diagrams below compact, we omit ⊗ symbols in nodesand arrows, and also omit parentheses in arrows. Proposition A.1.
We have that
Φ ( X ) := ( AX ) A is a right A -module in C .Proof. Compatibility of the right action with associativity follows from the com-mutativity of the diagram below. ((( AX ) A ) A ) A (1) α AXA,A,A / / α AX,A,A id A (cid:15) (cid:15) ((( AX ) A )( AA ) id AXA m A z z (2) ( AX )( A ( AA )) α − AX,A,AA ❤❤❤❤❤❤❤❤❤❤❤❤❤ id AX id A m A * * ❱❱❱❱❱❱❱❱❱❱❱❱❱ ( AX )(( AA ) A ) (4)(3) id AX α A,A,A O O id AX m A id A & & ( AX )( AA ) α − AX,A,A (cid:15) (cid:15) (( AX )( AA )) A α AX,AA,A ❤❤❤❤❤❤❤❤❤❤❤❤❤ id AX m A id A (cid:15) (cid:15) (( AX ) A ) A ρ AXA (cid:15) (cid:15) α AX,A,A / / ( AX )( AA ) id AX m A (5) t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ (( AX ) A ) A α AX,A,A t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ρ AXA i i (5) l l ( AX ) A ( AX )( AA ) id AX m A o o Here (1) is the pentagon axiom; (2) and (3) commute from the naturality of α ; (4)commutes from the associativity of m A ; and (5) is the definition of ρ .The compatibility of the right action with the right unitor holds from the fol-lowing diagram; we leave the compatibility of the right action with the left unitorto the reader. (( AX ) A ) α AX,A, (cid:15) (cid:15) id ( AX ) A u A / / (1) r AXA (3) * * (( AX ) A ) A α AX,A,A (cid:15) (cid:15) ( AX )( A ) id AX id A u A / / id AX r A (cid:15) (cid:15) ( AX )( AA ) (2) id AX m A r r ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ ( AX ) A Here, (1) commutes by the naturality of α ; (2) commutes from the unitality of thealgebra A ; and (3) commutes via the compatibility between α and r in C . (cid:3) Proposition A.2.
We have that
Φ ( X ) is an A -bimodule in C .Proof. The compatibility of left and right A -module structure is checked via thecommutativity of the diagram below. ( A (( AX ) A )) A α A,AXA,A / / α − A,AX,A id A (cid:15) (cid:15) (1) A ((( AX ) A ) A ) id A α AX,A,A (cid:15) (cid:15) (( A ( AX )) A ) A (2) α − A,A,X id AA (cid:15) (cid:15) α AAX,A,A + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲ A (( AX )( AA )) α − A,AX,AA s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣ id AAX m A (cid:15) (cid:15) (3) ((( AA ) X ) A ) A α AAX,A,A . . m A id XAA (cid:15) (cid:15) ( A ( AX ))( AA ) α − A,A,X id AA (cid:15) (cid:15) id AAX m A (5) (( AA ) X )( AA ) (6) id AAX m A ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ m A id XAA w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ A (( AX ) A ) α − A,AX,A (cid:15) (cid:15) (( AX ) A ) A (4) α AX,A,A (cid:15) (cid:15) ( A ( AX )) A α − A,A,X id A (cid:15) (cid:15) ( AX )( AA ) id AX m A / / ( AX ) A (( AA ) X ) A m A id XA o o Here, (1) is the pentagon axiom; (2), (3) and (4) commutes due to naturality of α ;and (5) and (6) clearly commutes. (cid:3) Proposition A.3.
In the non-strict case, we have that e Φ X,X ′ ( ρ Φ( X ) ⊗ id Φ( X ′ ) ) = e Φ X,X ′ (id Φ( X ) ⊗ λ Φ( X ′ ) ) α Φ( X ) ,A, Φ( X ′ ) . Proof.
Consider the diagram below. ((( AX ) A ) A )(( AX ′ ) A ) α AXA,A,AX ′ A (cid:15) (cid:15) (1) α AX,A,A id AX ′ A / / (( AX )( AA ))(( AX ′ ) A ) (2)id AX m A id AX ′ A / / α − AXAA,AX ′ ,A (cid:15) (cid:15) (( AX ) A )(( AX ′ ) A ) α − AXA,AX ′ ,A (cid:15) (cid:15) (( AX ) A )( A (( AX ′ ) A )) id AXA α − A,AX ′ ,A (cid:15) (cid:15) ((( AX )( AA ))( AX ′ )) A id AX m A id AX ′ A / / α AX,AA,AX ′ id A (cid:15) (cid:15) (3) ((( AX ) A )( AX ′ )) A α AX,A,AX ′ id A (cid:15) (cid:15) (( AX ) A )(( A ( AX ′ )) A ) id AXA α − A,A,X ′ id A (cid:15) (cid:15) (( AX )(( AA )( AX ′ ))) A id AX m A id AX ′ A / / id AX α − AA,A,X ′ id A (cid:15) (cid:15) (4) (( AX )( A ( AX ′ ))) A id AX α − A,A,X ′ id A (cid:15) (cid:15) (( AX ) A )((( AA ) X ′ ) A ) id AXA m A id X ′ A (cid:15) (cid:15) (( AX )((( AA ) A ) X ′ )) A id AX α A,A,A id X ′ A (cid:15) (cid:15) id AX m A id AX ′ A / / (5) (( AX )(( AA ) X ′ )) A id AX m A id X ′ A (cid:15) (cid:15) (( AX ) A )(( AX ′ ) A ) α − AXA,AX ′ ,A (cid:15) (cid:15) (( AX )(( A ( AA )) X ′ )) A id AXA m A id X ′ A (cid:15) (cid:15) (( AX )( AX ′ )) A id AX ε A id X ′ A (cid:15) (cid:15) ((( AX ) A )( AX ′ )) A α AX,A,AX ′ id A (cid:15) (cid:15) (( AX )(( AA ) X ′ )) A id AX m A id X ′ A ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ (6) (( AX )( X ′ )) A id AX l X ′ id A (cid:15) (cid:15) (( AX )( A ( AX ′ ))) A id AX α − A,A,X ′ id A (cid:15) (cid:15) (( AX ) X ′ ) A α A,X,X ′ id A (cid:15) (cid:15) (( AX )(( AA ) X ′ )) A id AX m A id X ′ A (cid:15) (cid:15) ( A ( XX ′ )) A (( AX )( AX ′ )) A id AX ε A id X ′ A / / (( AX )( X ′ )) A id AX l X ′ id A / / (( AX ) X ′ ) A α A,X,X ′ id A O O Here, (2), (3) and (4) commute by naturality of the associators; (5) commutesbecause m A is associative; (6) is equality. We will check the commutativity of (1) LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 41 in the following diagram: ((( AX ) A ) A )(( AX ′ ) A ) α AXA,A,AX ′ A (cid:15) (cid:15) α AX,A,A id AX ′ A / / (7) (( AX )( AA ))(( AX ′ ) A ) α − AXAA,AX ′ ,A (cid:15) (cid:15) (( AX ) A )( A (( AX ′ ) A )) id AXA α − A,AX ′ ,A (cid:15) (cid:15) ((( AX )( AA ))( AX ′ )) A α AX,AA,AX ′ id A (cid:15) (cid:15) (( AX ) A )(( A ( AX ′ )) A ) id AXA α − A,A,X ′ id A (cid:15) (cid:15) (( AX )( A ( A ( AX ′ ))) A id AXA α − A,A,X ′ id A (cid:15) (cid:15) (8) (( AX )(( AA )( AX ′ ))) A id AX α − AA,A,X ′ id A (cid:15) (cid:15) id AX α A,A,AX ′ id A o o (( AX ) A )((( AA ) X ′ ) A ) id AXA m A id X ′ A (cid:15) (cid:15) (( AX )( A (( AA ) X ′ ))) A (9)id AXA m A id X ′ A (cid:15) (cid:15) id AX α − A,AA,X ′ id A * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ (( AX )((( AA ) A ) X ′ )) A id AX α A,A,A id X ′ id A (cid:15) (cid:15) (( AX ) A )(( AX ′ ) A ) α − AXA,AX ′ ,A (cid:15) (cid:15) (( AX )( A ( AX ′ ))) A rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr id AX α − A,A,X ′ id A * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ (10) (( AX )(( A ( AA )) X ′ )) A id AX id A m A id X ′ id A (cid:15) (cid:15) ((( AX ) A )( AX ′ )) A α AX,A,AX ′ id A (cid:15) (cid:15) (( AX )(( AA ) X ′ )) A ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ (( AX )( A ( AX ′ ))) A id AX α − A,A,X ′ id A (cid:15) (cid:15) (( AX )(( AA ) X ′ )) A where (8) is the pentagon axiom, (9) is naturality of α and (10) clearly commutes.Finally, we will examine (7) in the next diagram. ((( AX ) A ) A )(( AX ′ ) A ) α AXA,A,AX ′ A (cid:15) (cid:15) α AX,A,A id AX ′ A / / (( AX )( AA ))(( AX ′ ) A ) α − AXAA,AX ′ ,A (cid:15) (cid:15) (( AX ) A )( A (( AX ′ ) A )) (12)id AXA α − A,AX ′ ,A (cid:15) (cid:15) ((( AX )( AA ))( AX ′ )) A α − AX,A,A id AX ′ A t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ α AX,AA,AX ′ id A (cid:15) (cid:15) (( AX ) A )(( A ( AX ′ )) A ) (13) α − AXA,AAX ′ ,A * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ id AXA α − A,A,X ′ id A (cid:15) (cid:15) (((( AX ) A ) A )( AX ′ )) A α AXA,A,AX ′ id A (cid:15) (cid:15) (14) α − AXAA,AX ′ ,A e e ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ (11) (( AX ) A )((( AA ) X ′ ) A ) α − AXA,AAX ′ ,A * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ id AXA m A id X ′ A (cid:15) (cid:15) ((( AX ) A )( A ( AX ′ ))) A (15) id AXA α − A,A,X ′ id A α AX,A,AAX ′ id A * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ (( AX )(( AA )( AX ′ ))) A id AX α A,A,AX ′ id A (cid:15) (cid:15) (( AX ) A )(( AX ′ ) A ) α − AXA,AX ′ ,A (cid:15) (cid:15) ((( AX ) A )(( AA ) X ′ )) A (16)id AXA m A id X ′ A t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ (( AX )( A ( A ( AX ′ )))) A id AXA α − A,A,X ′ id A (cid:15) (cid:15) ((( AX ) A )( AX ′ )) A α AX,A,AX ′ id A * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ (( AX )( A (( AA ) X ′ ))) A id AXA m A id X ′ A t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ (( AX )( A ( AX ′ ))) A Here, (12) and (14) are the pentagon axiom; (15) and (16) commute from thenaturality of α ; and (11) and (13) clearly commute. (cid:3) Proposition A.4.
We have that Φ is an A -bimodule map in C . Proof.
For brevity, we show that Φ is a right A -module map in C , and leave therest of the proof to the reader. The following diagram commutes: AA (1) m A / / ∆ A id A (cid:15) (cid:15) A ∆ A (cid:15) (cid:15) ( AA ) A (2) α A,A,A / / r − A id AA (cid:15) (cid:15) A ( AA ) (3)id A m A / / r − A id AA (cid:15) (cid:15) AA r − A id A (cid:15) (cid:15) (( A ) A ) A α A ,A,A / / ( A )( AA ) id A m A / / ( A ) A because (1) is the Frobenius condition of A ; (2) is the naturality of α ; and (3)commutes because the maps are applied in different slots. (cid:3) Proposition A.5.
We have that e Φ X,X ′ is an A -bimodule map.Proof. Again for brevity, we show that Φ
X,X ′ is a right A -module map for C strict,and leave the remainder of the proof to the reader. In the diagram Φ( XX ′ ) A (1)(2) ρ Φ( XX ′ ) (cid:15) (cid:15) Φ X,X ′ id A / / e Φ X,X ′ id A + + ❲❲❲❲❲❲❲❲❲❲❲❲❲ (Φ( X ) ⊗ A Φ( X ′ )) A ρ Φ( X ) ⊗ A Φ( X ′ ) (cid:15) (cid:15) (3) Φ( X )Φ( X ′ ) A π Φ( X ) , Φ( X ′ ) id A ❡❡❡❡❡❡❡❡❡❡❡❡❡ id Φ( X ) ρ Φ( X ′ ) (cid:15) (cid:15) Φ( X )Φ( X ′ ) π Φ( X ) , Φ( X ′ ) , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ Φ( XX ′ ) Φ X,X ′ / / e Φ X,X ′ ❣❣❣❣❣❣❣❣❣❣❣❣❣❣ (4) Φ( X ) ⊗ A Φ( X ′ ) , we have that (1) and (4) commute by the definition of Φ X,X ′ , and (3) commutes bythe definition of ρ Φ( X ) ⊗ A Φ( X ′ ) . Finally, (2) commutes because in the the followingdiagram each square commutes as the maps are applied in different slots. AXX ′ AA id AXX ′ m A (cid:15) (cid:15) id AX u A id X ′ AA / / AXAX ′ AA id AXAX ′ m A (cid:15) (cid:15) id AX ∆ A id X ′ AA / / AXAAX ′ AA id AXAAX ′ m A (cid:15) (cid:15) AXX ′ A id AX u A id X ′ A / / AXAX ′ A id AX ∆ A id X ′ A / / AXAAX ′ A (cid:3) Proposition A.6.
We have that Φ is an A -bimodule map.Proof. Recall that Φ = m A ( r A ⊗ id A ) : ( A ⊗ ) ⊗ A → A . We have that Φ isboth a left and a right A -module map for C strict due to the associativity of m A ;we leave the non-strict case to the reader. (cid:3) Proposition A.7.
We get that Φ ∗ , ∗ satisfies the associativity condition. LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 43
Proof.
Let us see the associativity condition holds for C strict, and we leave thenon-strict case for the reader. In fact, in the following diagram (Φ( X ) ⊗ A Φ( X ′ )) ⊗ A Φ( X ′′ ) (1)(2) Φ X,X ′ ⊗ A id Φ( X ′′ ) / / α A Φ( X ) , Φ( X ′ ) , Φ( X ′′ ) (cid:15) (cid:15) (6) Φ( XX ′ ) ⊗ A Φ( X ′′ ) Φ XX ′ ,X ′′ (cid:15) (cid:15) (7) (Φ( X ) ⊗ A Φ( X ′ ))Φ( X ′′ ) π Φ( X ) ⊗ A Φ( X ′ ) , Φ( X ′′ ) k k ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ Φ X,X ′ id Φ( X ′′ ) / / Φ( XX ′ )Φ( X ′′ ) π Φ( XX ′ ) , Φ( X ′′ ) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ e Φ XX ′ ,X ′′ (cid:25) (cid:25) ✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹ Φ( X ) ⊗ A (Φ( X ′ ) ⊗ A Φ( X ′′ )) (5)(3)id Φ( X ) ⊗ A Φ X ′ ,X ′′ (cid:15) (cid:15) Φ( X )Φ( X ′ )Φ( X ′′ ) id Φ( X ) π Φ( X ′ ) , Φ( X ′′ ) (cid:15) (cid:15) π Φ( X ) , Φ( X ′ ) id Φ( X ′′ ) O O e Φ X,X ′ id Φ( X ′′ ) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ id Φ( X ) e Φ X ′ ,X ′′ v v (8) Φ( X )(Φ( X ′ ) ⊗ A Φ( X ′′ )) π Φ( X ) , Φ( X ′ ) ⊗ A Φ( X ′′ ) c c id Φ( X ) Φ X,X ′ (cid:15) (cid:15) Φ( X )Φ( X ′ X ′′ ) π Φ( X ) , Φ( X ′ X ′′ ) s s ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ e Φ X,X ′ X ′′ , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ Φ( X ) ⊗ A Φ( X ′ X ′′ ) Φ X,X ′ X ′′ / / (4) Φ( XX ′ X ′′ ) , we have that (1) and (3) commute from (2.21); (2) commutes from the definition ofthe associativity constraint α A ; (4), (5), (6) and (7) commute from the definitionof Φ ∗ , ∗ . Moreover, (8) is the following diagram: AXAAX ′ AAX ′′ A id AXAAX ′ m A id X ′′ A (cid:15) (cid:15) id AX m A id X ′ AAX ′′ A / / AXAX ′ AAX ′′ A id AXAX ′ m A id X ′ A (cid:15) (cid:15) id AX ε A id X ′ AAX ′′ A / / AXX ′ AAX ′′ A id AXX ′ m A id X ′′ A (cid:15) (cid:15) AXAAX ′ AX ′′ A id AX m A id X ′ AX ′′ A / / id AXAAX ′ ε A id X ′′ A (cid:15) (cid:15) AXAX ′ AX ′′ A id AXAX ′ ε A id X ′′ A (cid:15) (cid:15) id AX ε A id X ′ AX ′′ A / / AXX ′ AX ′′ A id AXX ′ ε A id X ′′ A (cid:15) (cid:15) AXAAX ′ X ′′ A id AX m A id X ′ X ′′ A / / AXAX ′ X ′′ A id AX ε A id X ′ X ′′ A / / AXX ′ X ′′ A where each square commutes because the maps are applied in different slots. (cid:3) Proposition A.8.
We have that Φ satisfies the unitality condition.Proof. We verify the left unitality condition when C is strict, and leave the non-strict case and the rest to the reader. In the diagram A ⊗ A Φ( X ) l A Φ( X ) (cid:15) (cid:15) Φ ⊗ A id Φ( X ) / / Φ( ) ⊗ A Φ( X ) Φ ,X (cid:15) (cid:15) A Φ( X ) (1) (2)(3)Φ id Φ( X ) / / λ Φ( X ) u u ❥❥❥❥❥❥❥❥❥❥❥❥ π A, Φ( X ) i i ❚❚❚❚❚❚❚❚❚❚ Φ( )Φ( X ) (4) e Φ ,X * * ❯❯❯❯❯❯❯❯❯❯❯❯❯ π Φ( ) , Φ( X ) ✐✐✐✐✐✐✐✐✐✐✐ Φ( X ) Φ( X ) , We have that (1) is commutative from the definition of l A Φ( X ) ; (3) is commutativeby (2.21); and (4) is commutative by the definition of Φ ,X . The diagram (2) is the following: AAXA (5) m A id XA (cid:15) (cid:15) ∆ A id AXA / / AAAXA id A m A id XA (cid:15) (cid:15) AAXA id A ε A id XA (cid:15) (cid:15) AXA ∆ A id XA ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ AXA (6) where (5) commutes from the Frobenius compatibility between m A and ∆ A , and (6)commutes from the counit axiom of A . (cid:3) Proposition A.9.
We get that Φ ∗ , ∗ satisfies the coassociativity condition.Proof. Let us prove the coassociativity constraint for C strict, and we omit thenon-strict case here. In the diagram below: Φ( XX ′ X ′′ ) Φ X,X ′ X ′′ / / Φ XX ′ ,X ′′ (cid:15) (cid:15) e Φ XX ′ ,X ′′ (cid:25) (cid:25) ✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹ e Φ X,X ′ X ′′ , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ (1)(8) (6) Φ( X ) ⊗ A Φ( X ′ X ′′ ) id Φ( X ) ⊗ A Φ X ′ ,X ′′ (cid:15) (cid:15) (2) Φ( X )Φ( X ′ X ′′ ) id Φ( X ) Φ X ′ ,X ′′ (cid:15) (cid:15) π Φ( X ) , Φ( X ′ X ′′ ) ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ id Φ( X ) e Φ X ′ ,X ′′ ) ) Φ( X )(Φ( X ′ ) ⊗ A Φ( X ′′ )) π Φ( X ) , Φ( X ′ ) ⊗ A Φ( X ′′ ) & & Φ( X )Φ( X ′ )Φ( X ′′ ) id Φ( X ) π Φ( X ′ ) , Φ( X ′′ ) O O π Φ( X ) , Φ( X ′ ) id Φ( X ′′ ) (cid:15) (cid:15) Φ( X ) ⊗ A (Φ( X ′ ) ⊗ A Φ( X ′′ )) (3) Φ( XX ′ )Φ( X ′′ ) π Φ( XX ′ ) , Φ( X ′′ ) u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ e Φ X,X ′ id Φ( X ′′ ) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Φ X,X ′ id Φ( X ′′ ) / / (Φ( X ) ⊗ A Φ( X ′ ))Φ( X ′′ ) π Φ( X ) ⊗ A Φ( X ′ ) , Φ( X ′′ ) + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ Φ( XX ′ ) ⊗ A Φ( X ′′ ) (7) Φ X,X ′ ⊗ A id Φ( X ′′ ) / / (Φ( X ) ⊗ A Φ( X ′ )) ⊗ A Φ( X ′′ ) , α A Φ( X ) , Φ( X ′ ) , Φ( X ′′ ) O O (4)(5) We have that (2) and (4) commute from (2.21); (3) commutes from the definitionof the associativity constraint α A ; (1), (5), (6) and (7) commute from the definitionof Φ ∗ , ∗ ; and (8) is the following diagram, AXX ′ X ′′ A id AXX ′ u A id X ′ A (cid:15) (cid:15) id AX u A id X ′ X ′′ A / / AXAX ′ X ′′ A id AXAX ′ u A id X ′′ A (cid:15) (cid:15) id AX ∆ A id X ′ X ′′ A / / AXAAX ′ X ′′ A id AXAAX ′ u A id X ′′ A (cid:15) (cid:15) AXX ′ AX ′′ A id AX u A id X ′ AX ′′ A / / id AXX ′ ∆ A id X ′′ A (cid:15) (cid:15) AXAX ′ AX ′′ A id AXAX ′ ∆ A id X ′′ A (cid:15) (cid:15) id AX ∆ A id X ′ AX ′′ A / / AXAAX ′ AX ′′ A id AXAAX ′ ∆ A id X ′′ A (cid:15) (cid:15) AXX ′ AAX ′′ A id AX u A id X ′ AAX ′′ A / / AXAX ′ AAX ′′ A id AX ∆ A id X ′ AAX ′′ A / / AXAAX ′ AAX ′′ A where each square commutes because the maps are applied in different slots. (cid:3) Appendix B. Remainder of the proof of Theorem 4.1
In this appendix, we fill in some details for the proof of Theorem 4.1.
LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 45
Proposition B.1.
We have that ( P, λ Γ( S ) P , ρ Γ( S ′ ) P ) ∈ Γ( S ) T Γ( S ′ ) , ( Q, λ Γ( S ′ ) Q , ρ Γ( S ) Q ) ∈ Γ( S ′ ) T Γ( S ) , where λ Γ( S ) P = Γ( λ SP ) Γ S,P : Γ( S ) ⊗ T P → P, ρ Γ( S ′ ) P = Γ( ρ S ′ P ) Γ P,S ′ : P ⊗ T Γ( S ′ ) → P,λ Γ( S ′ ) Q = Γ( λ S ′ Q ) Γ S ′ ,Q : Γ( S ′ ) ⊗ T Q → Q, ρ Γ( S ) Q = Γ( ρ SQ ) Γ Q,S : Q ⊗ T Γ( S ) → Q. Proof.
It is straight-forward to check that P is a right Γ( S ′ )-module in T withaction given by ρ Γ( S ′ ) P . In a similar way, it can be seen that P is a left Γ( S )-modulein T with action λ Γ( S ) P . Let us now check the left and right action compatibilityfor P . Consider the diagram, where ⊗ := ⊗ S and we suppress the ⊗ ∗ symbols inmorphisms below. (Γ( S ) ⊗ T P ) ⊗ T Γ( S ′ ) α Γ( S ) ,P, Γ( S ′ ) / / λ Γ( S ) P id (cid:15) (cid:15) Γ S,P id ( ( ❘❘❘❘❘❘❘❘❘❘❘ Γ( S ) ⊗ T ( P ⊗ T Γ( S ′ )) id ρ Γ( S ′ ) P (cid:15) (cid:15) id Γ P,S ′ v v ❧❧❧❧❧❧❧❧❧❧❧ Γ( S ⊗ P ) ⊗ T Γ( S ′ ) Γ( λSP )id | | ③③③③③③③③③③③③③③③③③③ Γ S ⊗ P,S ′ (cid:15) (cid:15) (1) Γ( S ) ⊗ T Γ( P ⊗ S ′ ) id Γ( ρS ′ P ) " " ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ Γ S,P ⊗ S (cid:15) (cid:15) Γ(( S ⊗ P ) ⊗ S ′ ) Γ( λSP id)(3) (cid:15) (cid:15) Γ( αS,P,S ′ ) / / Γ( S ⊗ ( P ⊗ S ′ )) Γ(id ρS ′ P ) (4) (cid:15) (cid:15) P ⊗ T Γ( S ′ ) ρ Γ( S ′ ) P + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ Γ P,S ′ / / Γ( P ⊗ S ′ ) Γ( ρS ′ P ) $ $ ❏❏❏❏❏❏❏❏ (2) Γ( S ⊗ P ) Γ( λSP ) z z tttttttt Γ( S ) ⊗ T P λ Γ( S ) P s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ Γ S,P o o P Here, (1) commutes as Γ is a monoidal functor, and (2) commutes since P ∈ S C S ′ .The diagrams (3) and (4) commute due to the naturality of Γ ∗ , ∗ , and the trianglescorrespond to the definition of the left and right actions of P in Γ( S ) T Γ( S ′ ) . There-fore, ( P, λ Γ( S ) P , ρ Γ( S ′ ) P ) ∈ Γ( S ) T Γ( S ′ ) . Analogously, ( Q, λ Γ( S ′ ) Q , ρ Γ( S ) Q ) ∈ Γ( S ′ ) T Γ( S ) . (cid:3) Proposition B.2.
The epimorphisms τ : P ⊗ Γ( S ′ ) Q ։ Γ( S ) ∈ Γ( S ) T Γ( S ) ,µ : Q ⊗ Γ( S ) P ։ Γ( S ′ ) ∈ Γ( S ′ ) T Γ( S ′ ) , satisfy diagrams ( ∗ ) and ( ∗∗ ) in Proposition 2.25(b).Proof. Diagram ( ∗ ) corresponds to the following; ⊗ is understood from context: (cid:2) P ⊗ Γ( S ′ ) Q (cid:3) ⊗ Γ( S ) P τ ⊗ Γ( S ) id P (cid:30) (cid:30) (7 ′ ) α P,Q,P / / P ⊗ Γ( S ′ ) (cid:2) Q ⊗ Γ( S ) P (cid:3) id P ⊗ Γ ( S ′ ) µ (cid:0) (cid:0) (7) ( P ⊗ Q ) ⊗ Γ( S ) P (5 ′ ) π Γ( S ′ ) P,Q ⊗ Γ( S ) id g g ❖❖❖❖❖❖❖❖❖❖❖ (1) P ⊗ Γ( S ′ ) ( Q ⊗ P ) id ⊗ Γ( S ′ ) π Γ( S ) Q,P ♦♦♦♦♦♦♦♦♦♦♦♦ (5) (cid:0) P ⊗ Γ( S ′ ) Q (cid:1) ⊗ P π Γ( S ) PQ,P O O τ ⊗ id P (cid:31) (cid:31) ( P ⊗ Q ) ⊗ P π Γ( S ) PQ,P g g ❖❖❖❖❖❖❖❖❖❖❖ α P,Q,P / / Γ P,Q ⊗ id w w ♣♣♣♣♣♣♣♣♣♣♣ π Γ( S ′ ) P,Q ⊗ id o o P ⊗ ( Q ⊗ P ) id ⊗ π Γ( S ) Q,P / / π Γ( S ′ ) P,QP ♦♦♦♦♦♦♦♦♦♦♦ id ⊗ Γ Q,P ' ' ◆◆◆◆◆◆◆◆◆◆◆ P ⊗ (cid:0) Q ⊗ Γ( S ) P (cid:1) π Γ( S ′ ) P,QP O O id P ⊗ µ (cid:127) (cid:127) (6 ′ ) Γ (cid:0) P ⊗ Q (cid:1) ⊗ P Γ( π S ′ P,Q ) ⊗ id (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Γ PQ,P ' ' ◆◆◆◆◆◆◆◆◆◆◆ (2) P ⊗ Γ (cid:0) Q ⊗ P (cid:1) id ⊗ Γ( π SQ,P ) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ Γ P,QP w w ♣♣♣♣♣♣♣♣♣♣♣ (6)(8 ′ ) Γ (cid:0)(cid:2) P ⊗ Q (cid:3) ⊗ P (cid:1) Γ( α P,Q,P ) $ $ Γ( π S ′ P,Q ⊗ id) (cid:15) (cid:15) Γ (cid:0) P ⊗ (cid:2) Q ⊗ P (cid:3)(cid:1) Γ(id ⊗ π SQ,P ) (cid:15) (cid:15) (8)Γ (cid:0) P ⊗ S ′ Q (cid:1) ⊗ P Γ PQ,P / / Γ( τ ) ⊗ id (cid:15) (cid:15) Γ (cid:0)(cid:2) P ⊗ S ′ Q (cid:3) ⊗ P (cid:1) Γ( τ ⊗ id) w w ♣♣♣♣♣♣♣♣♣♣♣ (10 ′ ) Γ( π SPQ,P ) (cid:15) (cid:15) (9 ′ ) (3) Γ (cid:0) P ⊗ (cid:2) Q ⊗ S P (cid:3)(cid:1) (10) (9)Γ( π S ′ P,QP ) (cid:15) (cid:15) Γ(id ⊗ µ ) ' ' ◆◆◆◆◆◆◆◆◆◆◆ P ⊗ Γ (cid:0) Q ⊗ S P (cid:1) id ⊗ Γ( µ ) (cid:15) (cid:15) Γ P,QP o o Γ ( S ) ⊗ P π Γ( S )Γ( S ) ,P (cid:15) (cid:15) Γ S,P / / Γ (cid:0) S ⊗ P (cid:1) Γ( π SS,P ) (cid:15) (cid:15) Γ (cid:0)(cid:2) P ⊗ S ′ Q (cid:3) ⊗ S P (cid:1) Γ( α P,Q,P ) ; ; Γ( τ ⊗ S id) w w ♣♣♣♣♣♣♣♣♣♣♣ Γ (cid:0) P ⊗ S ′ (cid:2) Q ⊗ S P (cid:3)(cid:1) Γ(id ⊗ S ′ µ ) ' ' ◆◆◆◆◆◆◆◆◆◆◆ Γ (cid:0) P ⊗ S ′ (cid:1) Γ( π S ′ P,S ′ ) (cid:15) (cid:15) P ⊗ Γ ( S ′ ) π Γ( S ′ ) P, Γ( S ′ ) (cid:15) (cid:15) Γ P,S ′ o o Γ (cid:0) S ⊗ S P (cid:1) (11 ′ ) Γ( l SP ) * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ (4) Γ (cid:0) P ⊗ S ′ S ′ (cid:1) (11)Γ( r S ′ P ) t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ Γ ( S ) ⊗ Γ( S ) P l Γ( S ) P / / P P ⊗ Γ( S ′ ) Γ ( S ′ ) r Γ( S ′ ) P o o Diagram (1) is the definition of α (see Definition 2.23). Diagram (2) commutesas Γ is a monoidal functor, and (3) results from applying Γ to the definition of α .Diagram (4) is the result of applying the functor Γ to the diagram (*). Diagrams (5)and (7) follow from (2.21). Diagram (6) is (4.2). Diagrams (8) and (9) commutefrom naturality of Γ ∗ , ∗ . Diagram (10) commutes by applying Γ to (2.21). Theproof of diagram (11) is given below. Finally, the commutativity of (5 ′ )–(11 ′ )follow analogously to the proof of (5)–(11), respectively. Therefore, diagram ( ∗ )commutes. In an analogous manner, diagram ( ∗∗ ) commutes. Γ( P ) ⊗ T Γ( S ′ ) Γ P,S ′ / / π Γ( S ′ )Γ( P ) , Γ( S ′ ) (cid:15) (cid:15) ρ Γ( S ′ )Γ( P ) (cid:25) (cid:25) Γ( P ⊗ S S ′ ) Γ( π S ′ P,S ′ ) (cid:15) (cid:15) Γ( ρ S ′ P ) (cid:2) (cid:2) Γ( P ) ⊗ Γ( S ′ ) Γ( S ′ ) r Γ( S ′ )Γ( P ) * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ (def. of r Γ( S ′ )Γ( P ) ) (def. of ρ Γ( S ′ ) P ) (def. of r S ′ P ) Γ( P ⊗ S ′ S ′ ) Γ( r S ′ P ) u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Γ( P ) (cid:3) Appendix C. Direct proof that A K,β ( L, ψ ) is Frobenius In Section 6, we proved that the object A K,β ( L, ψ ) in the category C ( G, ω, K, β )is a Frobenius algebra using a functorial approach [Theorem 6.4]. In this appendix,we will give a direct proof of this result. Recall the structure maps m , u , ∆, ε of A K,β ( L, ψ ) in the statement of Theorem 6.4, recall the associativity isomorphism of
LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 47 C ( G, ω, K, β ) from Proposition 2.20 and Definition 5.2, and let A denote A ( K, β ).The result will hold via Propositions C.1–C.6 below.
Proposition C.1.
The maps m , u , ∆ , ε are A ( K, β ) -bimodule maps in Vec ωG .Proof. Take A := A ( K, β ) and B := A ( L, ψ ), and let
ABA := A K,β ( L, ψ ). First,it is clear that each of the maps m , u , ∆, ε is a morphism in Vec ωG . Next, we verifythat they are indeed A -bimodule maps.The multiplication map m ABA is a left A -module map due to the followingcalculation: e m ABA λ AABA ⊗ A ABA [ δ d ′ ⊗ ((( δ f ⊗ δ g ) ⊗ δ k ) ⊗ (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]= e m ABA ( m A ⊗ id B,A,ABA )( α − A,A,B ⊗ id A,ABA )( α − A,AB,A ⊗ id ABA ) ◦ α − A,ABA,ABA [ δ d ′ ⊗ ((( δ f ⊗ δ g ) ⊗ δ k ) ⊗ (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]= ω ( d ′ , f gk, f ′ g ′ k ′ ) ω ( d ′ , f g, k ) ω ( d ′ , f, g ) β ( d ′ , f ) m ABA [(( δ d ′ f ⊗ δ g ) ⊗ δ k ) ⊗ (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]= δ kf ′ ,e ω ( d ′ , f gk, f ′ g ′ k ′ ) ω ( d ′ , f g, k ) ω ( d ′ , f, g ) ω ( d ′ f gk, f ′ g ′ , k ′ ) · ω − ( d ′ f g, k, f ′ g ′ ) ω − ( d ′ f, g, g ′ ) ω ( k, f ′ , g ′ ) ψ ( g, g ′ ) · β ( d ′ , f ) β ( k, f ′ )[( δ d ′ f ⊗ δ gg ′ ) ⊗ δ k ′ ]= δ kf ′ ,e ω ( d ′ , f, gg ′ ) ω ( d ′ , f gg ′ , k ′ ) ω − ( f, g, g ′ ) ω − ( f g, k, f ′ g ′ ) · ω ( f gk, f ′ g ′ , k ′ ) ω ( k, f ′ , g ′ ) ψ ( g, g ′ ) β ( d ′ , f ) β ( k, f ′ )[( δ d ′ f ⊗ δ gg ′ ) ⊗ δ k ′ ]= ( m A ⊗ id B ⊗ id A )( α − A,A,B ⊗ id A ) α − A,AB,A (id A ⊗ m ABA )[ δ d ′ ⊗ ((( δ f ⊗ δ g ) ⊗ δ k ) ⊗ (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]= λ AABA (id A ⊗ e m ABA )[ δ d ′ ⊗ ((( δ f ⊗ δ g ) ⊗ δ k ) ⊗ (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]where the fourth equality holds by (5.4) with ( g , g , g , g ) equal to ( d ′ , f, g, g ′ ),to ( d ′ , f gk, f ′ g ′ , k ′ ), and then to ( d ′ , f g, k, f ′ g ′ ). We leave it to the reader to checkthat m ABA is a right A -module map.The unit map u ABA is an A -bimodule map due to the following calculations: λ AABA (id A ⊗ u ABA )[ δ d ′ ⊗ δ d ]= L s ∈ K ( m A ⊗ id B ⊗ id A )( α − A,A,B ⊗ id A ) α − A,AB,A · β − ( ds − , s )[ δ d ′ ⊗ (( δ ds − ⊗ δ e ) ⊗ δ s )]= L s ∈ K ω ( d ′ , ds − , s ) β ( d ′ , ds − ) β − ( ds − , s )[( δ d ′ ds − ⊗ δ e ) ⊗ δ s ]= L s ∈ K β ( d ′ , d ) β − ( d ′ ds − , s )[( δ d ′ ds − ⊗ δ e ) ⊗ δ s ]= β ( d ′ , d ) u ABA [ δ d ′ d ]= u ABA m A [ δ d ′ ⊗ δ d ] , where the third equality holds by (5.6) with θ = β and ( f , f , f ) = ( d ′ , ds − , s );and ρ AABA ( u ABA ⊗ id A )[ δ d ⊗ δ d ′′ ] = L s ∈ K β − ( ds − , s ) (id A ⊗ id B ⊗ m A ) α AA,B,A [(( δ ds − ⊗ δ e ) ⊗ δ s ) ⊗ δ d ′′ ]= L s ∈ K ω − ( ds − , s, d ′′ ) β ( s, d ′′ ) β − ( ds − , s )[( δ ds − ⊗ δ e ) ⊗ δ sd ′′ ]= L s ∈ K β ( d, d ′′ ) β − ( ds − , sd ′′ )[( δ ds − ⊗ δ e ) ⊗ δ sd ′′ ] t = sd ′′ = L t ∈ K β ( d, d ′′ ) β − ( dd ′′ t − , t )[( δ dd ′′ t − ⊗ δ e ) ⊗ δ t ]= β ( d, d ′′ ) u ABA [ δ dd ′′ ]= u ABA m A [ δ d ⊗ δ d ′′ ] , where the third equality holds by (5.6) with θ = β and ( f , f , f ) = ( ds − , s, d ′′ ) . The comultiplication map ∆
ABA is a left A -module map due to the followingcalculation. Since λ AABA ⊗ A ABA (id A ⊗ π ABA,ABA ) = π ABA,ABA ( λ AABA ⊗ id ABA ) α − A,ABA,ABA , we have to see that ( λ AABA ⊗ id ABA ) α − A,ABA,ABA (id A ⊗ ∆ ′ ABA ) = ∆ ′ ABA λ AABA , for π AABA,ABA ∆ ′ ABA = ∆
ABA . In fact,( λ AABA ⊗ id ABA ) α − A,ABA,ABA (id A ⊗ ∆ ′ ABA )[ δ d ′ ⊗ (( δ f ⊗ δ g ) ⊗ δ k ]= ( m A ⊗ id B,A,A,B,A )( α − A,A,B ⊗ id A,A,B,A )( α − A,AB,A ⊗ id A,B,A ) α − A,ABA,ABA ◦ (id A ⊗ ∆ ′ ABA )[ δ d ′ ⊗ (( δ f ⊗ δ g ) ⊗ δ k ]= | K | − | L | − L h ∈ L ; s ∈ K ω ( f, gh, h − ) ω ( f gh, s, s − h − ) ω − ( f ghs, s − h − , k ) · ω − ( s, s − , h − ) ψ − ( gh, h − ) β − ( s, s − ) · ( m A ⊗ id B,A,A,B,A )( α − A,A,B ⊗ id A,A,B,A )( α − A,AB,A ⊗ id A,B,A ) ◦ α − A,ABA,ABA [ δ d ′ ⊗ ((( δ f ⊗ δ gh ) ⊗ δ s ) ⊗ (( δ s − ⊗ δ h − ) ⊗ δ k ))]= | K | − | L | − L h ∈ L ; s ∈ K ω ( f, gh, h − ) ω ( f gh, s, s − h − ) ω − ( f ghs, s − h − , k ) · ω − ( s, s − , h − ) ω ( d ′ , f ghs, s − h − k ) ω ( d ′ , f gh, s ) ω ( d ′ , f, gh ) · ψ − ( gh, h − ) β − ( s, s − ) β ( d ′ , f )[( δ d ′ f ⊗ δ gh ) ⊗ δ s ) ⊗ (( δ s − ⊗ δ h − ) ⊗ δ k ))]= | K | − | L | − L h ∈ L ; s ∈ K ω ( d ′ f, gh, h − ) ω ( d ′ f gh, s, s − h − ) · ω − ( d ′ f ghs, s − h − , k ) ω − ( s, s − , h − ) ω ( d ′ , f g, k ) ω ( d ′ , f, g ) · ψ − ( gh, h − ) β − ( s, s − ) β ( d ′ , f )[( δ d ′ f ⊗ δ gh ) ⊗ δ s ) ⊗ (( δ s − ⊗ δ h − ) ⊗ δ k ))]= ω ( d ′ , f g, k ) ω ( d ′ , f, g ) β ( d ′ , f )∆ ′ ABA [( δ d ′ f ⊗ δ g ) ⊗ δ k ]= ∆ ′ ABA ( m A ⊗ id B ⊗ id A )( α − A,A,B ⊗ id A ) α − A,AB,A [ δ d ′ ⊗ (( δ f ⊗ δ g ) ⊗ δ k ]= ∆ ′ ABA λ AABA [ δ d ′ ⊗ (( δ f ⊗ δ g ) ⊗ δ k ] , where the fourth equality holds by (5.4) with ( g , g , g , g ) substituted for( d ′ , f ghs, s − h − , k ), for ( d ′ , f gh, s, s − h − ), and then for ( d ′ , f, gh, h − ). We leaveit to the reader to check that ∆ ABA is a right A -module map. LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 49
The counit map ε ABA is an A -bimodule map due to the following calculations: m A (id A ⊗ ε ABA )[ δ d ′ ⊗ (( δ f ⊗ δ g ) ⊗ δ k )]= | L | δ g,e β ( f, k ) β ( d ′ , f k ) δ d ′ fk = | L | δ g,e ω ( d ′ , f, k ) β ( d ′ , f ) β ( d ′ f, k ) δ d ′ fk = ω ( d ′ , f g, k ) ω ( d ′ , f, g ) β ( d ′ , f ) ε ABA [(( δ d ′ f ⊗ δ g ) ⊗ δ k ]= ε ABA ( m A ⊗ id B ⊗ id A )( α − A,A,B ⊗ id A ) α − A,AB,A [ δ d ′ ⊗ (( δ f ⊗ δ g ) ⊗ δ k )]= ε ABA λ AABA [ δ d ′ ⊗ (( δ f ⊗ δ g ) ⊗ δ k )] , where the second equality holds by (5.6) with θ = β and ( f , f , f ) = ( d ′ , f, k );and m A ( ε ABA ⊗ id A )[(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ δ d ′′ ]= | L | δ g,e β ( f, k ) β ( f k, d ′′ ) δ fkd ′′ = | L | δ g,e ω − ( f, k, d ′′ ) β ( k, d ′′ ) β ( f, kd ′′ ) δ fkd ′′ = ω − ( f g, k, d ′′ ) β ( k, d ′′ ) ε ABA [(( δ f ⊗ δ g ) ⊗ δ kd ′′ ]= ε ABA (id A ⊗ id B ⊗ m A ) α AA,B,A [(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ δ d ′′ ]= ε ABA ρ AABA [(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ δ d ′′ ] , where the second equality holds by (5.6) with θ = β and ( f , f , f ) = ( f, k, d ′′ ). (cid:3) Proposition C.2.
We have that m is associative.Proof. Without loss of generality, assume that m is nonzero. On one side, takinginto account the change of variable f ′ = k − due to the term δ kf ′ ,e and f ′′ = k ′− due to the term δ k ′ f ′′ ,e , we get m ( m ⊗ A id)[[(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )] ⊗ A (( δ f ′′ ⊗ δ g ′′ ) ⊗ δ k ′′ )]= ω ( f gk, k − g ′ , k ′ ) ω − ( f g, k, k − g ′ ) ω ( k, k − , g ′ ) ω − ( f, g, g ′ ) · β ( k, k − ) ψ ( g, g ′ ) m [( δ f ⊗ δ gg ′ ⊗ δ k ′ ) ⊗ A ( δ f ′′ ⊗ δ g ′′ ⊗ δ k ′′ )]= ω ( f gk, k − g ′ , k ′ ) ω − ( f g, k, k − g ′ ) ω ( k, k − , g ′ ) ω − ( f, g, g ′ ) · ω ( f gg ′ k ′ , k ′− g ′′ , k ′′ ) ω − ( f gg ′ , k ′ , k ′− g ′′ ) ω ( k ′ , k ′− , g ′′ ) ω − ( f, gg ′ , g ′′ ) · β ( k, k − ) β ( k ′ , k ′− ) ψ ( g, g ′ ) ψ ( gg ′ , g ′′ )[ δ f ⊗ δ gg ′ g ′′ ⊗ δ k ′′ ] . On the other hand, m (id ⊗ A m ) α [[(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )] ⊗ A (( δ f ′′ ⊗ δ g ′′ ) ⊗ δ k ′′ )]= ω − ( f gk, k − g ′ k ′ , k ′− g ′′ k ′′ ) ω ( k − g ′ k ′ , k ′− g ′′ , k ′′ ) ω − ( k − g ′ , k ′ , k ′− g ′′ ) · ω ( k ′ , k ′− , g ′′ ) ω − ( k − , g ′ , g ′′ ) β ( k ′ , k ′− ) ψ ( g ′ , g ′′ ) · m [( δ f ⊗ δ g ⊗ δ k ) ⊗ A ( δ f ′ ⊗ δ g ′ g ′′ ⊗ δ k ′′ ) ] . = ω − ( f gk, k − g ′ k ′ , k ′− g ′′ k ′′ ) ω ( k − g ′ k ′ , k ′− g ′′ , k ′′ ) ω − ( k − g ′ , k ′ , k ′− g ′′ ) · ω ( k ′ , k ′− , g ′′ ) ω − ( k − , g ′ , g ′′ ) ω ( f gk, k − g ′ g ′′ , k ′′ ) ω − ( f g, k, k − g ′ g ′′ ) · ω ( k, k − , g ′ g ′′ ) ω − ( f, g, g ′ g ′′ ) β ( k ′ , k ′− ) β ( k, k − ) ψ ( g ′ , g ′′ ) ψ ( g, g ′ g ′′ ) · [ δ f ⊗ δ gg ′ g ′′ ⊗ δ k ′′ ]The two sides are equal after applying (5.6) with θ = ψ and ( f , f , f ) = ( g, g ′ , g ′′ )and then applying (5.4) with the values of the tuple ( g , g , g , g ) assigned sequen-tially as follows: ( f, g, g ′ , g ′′ ); ( f gk, k − g ′ k ′ , k ′− g ′′ , k ′′ ); ( f gk, k − g ′ , k ′ , k ′− g ′′ );( k, k − , g ′ , g ′′ ); ( f g, k, k − g ′ , g ′′ ) . (cid:3) Proposition C.3.
We have that u satisfies the unit axiom with respect to m .Proof. First, let us check the right unitality condition, m (id ⊗ A u ) = r AA K,β ( L,ψ ) .By applying the left hand side to [( δ f ⊗ δ g ) ⊗ δ k ] ⊗ A ( δ d ), we get m (cid:0)L s ∈ K β − ( ds − , s )[( δ f ⊗ δ g ) ⊗ δ k ] ⊗ A [( δ ds − ⊗ δ e ) ⊗ δ s ] (cid:1) = L s ∈ K δ kds − ,e ω ( f gk, ds − , s ) ω − ( f g, k, ds − ) β − ( ds − , s ) β ( k, ds − ) · [ δ f ⊗ δ g ⊗ δ s ]= ω ( f gk, k − , kd ) ω − ( f g, k, k − ) β − ( k − , kd ) β ( k, k − )[ δ f ⊗ δ g ⊗ δ kd ]= β ( k, d ) ω − ( f g, k, d )[( δ f ⊗ δ g ) ⊗ δ kd ]= r AA K,β ( L,ψ ) ([( δ f ⊗ δ g ) ⊗ δ k ] ⊗ A ( δ d )) , where the third equation holds by using (5.4) with ( g , g , g , g ) = ( f g, k, k − , kd ),and then using (5.6) with θ = β and ( f , f , f ) = ( k, k − , kd ).For the left unitality condition, m ( u ⊗ A id) = l AA K,β ( L,ψ ) , we have: m ( u ⊗ A id)[ δ d ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]= m (cid:0)L s ∈ K β − ( ds − , s )[(( δ ds − ⊗ δ e ) ⊗ δ s ) ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )] (cid:1) = ω ( d, f ′ g ′ , k ′ ) ω ( f ′− , f ′ , g ′ ) ω − ( df ′ , f ′− , f ′ g ′ ) β − ( df ′ , f ′− ) β ( f ′− , f ′ ) · [( δ df ′ ⊗ δ g ′ ) ⊗ δ k ′ ]= β ( d, f ′ ) ω ( d, f ′ g ′ , k ′ ) ω ( d, f ′ , g ′ )[( δ df ′ ⊗ δ g ′ ) ⊗ δ k ′ ]= l AA K,β ( L,ψ ) ([ δ d ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]) , where the second equation holds by the substitution s = f ′− due to the term δ sf ′ ,e in the formula for m . Moreover, the third equality holds by applying (5.6) with θ = β with ( f , f , f ) = ( f ′ , f ′− , f ′ ) and ( d, f ′ , f ′− ), and by applying (5.4) with( g , g , g , g ) = ( f ′ , f ′− , f ′ , g ′ ) and ( d, f ′ , f ′− , f ′ g ′ ). (cid:3) Proposition C.4.
We get that ∆ is coassociative.Proof. On the one hand, we have α (∆ ⊗ A id)∆(( δ f ⊗ δ g ) ⊗ δ k )= | K | − | L | − α (∆ ⊗ A id) (cid:18) L h ∈ L,s ∈ K ω ( f, gh, h − ) ω ( f gh, s, s − h − ) · ω − ( f gh, s, s − h − , k ) ω − ( s, s − , h − ) ψ − ( gh, h − ) β − ( s, s − ) (cid:19) · [(( δ f ⊗ δ gh ) ⊗ δ s ) ⊗ A (( δ s − ⊗ δ h − ) ⊗ δ k )] LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 51 = | K | − | L | − L h,l ∈ L ; s,t ∈ K ω ( f, gh, h − ) ω ( f, ghl, l − ) ω ( f ghl, t, t − l − ) · ω ( f gh, s, s − h − ) ω − ( f ghs, s − h − , k ) ω − ( s, s − , h − ) ω − ( f ghlt, t − l − , s ) · ω − ( t, t − , l − ) ψ − ( gh, h − ) ψ − ( ghl, l − ) β − ( s, s − ) β − ( t, t − ) · α [(( δ f ⊗ δ ghl ) ⊗ δ t ) ⊗ A [(( δ t − ⊗ δ l − ) ⊗ δ s ) ⊗ A (( δ s − ⊗ δ h − ) ⊗ δ k )]]= | K | − | L | − L h,l ∈ L ; s,t ∈ K ω ( f, gh, h − ) ω ( f, ghl, l − ) ω ( f ghl, t, t − l − ) · ω ( f gh, s, s − h − ) ω − ( f ghlt, t − l − s, s − h − k ) ω − ( f ghs, s − h − , k ) · ω − ( s, s − , h − ) ω − ( t, t − , l − ) ω − ( f ghlt, t − l − , s ) · ψ − ( gh, h − ) ψ − ( ghl, l − ) β − ( s, s − ) β − ( t, t − ) · (( δ f ⊗ δ ghl ) ⊗ δ t ) ⊗ A [(( δ t − ⊗ δ l − ) ⊗ δ s ) ⊗ A (( δ s − ⊗ δ h − ) ⊗ δ k )] . On the other hand, we get(id ⊗ A ∆)∆(( δ f ⊗ δ g ) ⊗ δ k )= | K | − | L | − (id ⊗ A ∆) (cid:18) L h ∈ L,s ∈ K ω ( f, gh, h − ) ω ( f gh, s, s − h − ) · ω − ( f ghs, s − h − , k ) ω − ( s, s − , h − ) ψ − ( gh, h − ) β − ( s, s − ) · [( δ f ⊗ δ gh ) ⊗ δ s ] ⊗ A [( δ s − ⊗ δ h − ) ⊗ δ k ] (cid:19) = | K | − | L | − L h,l ∈ L ; s,t ∈ K ω ( f, gh, h − ) ω ( f gh, s, s − h − ) ω ( s − , h − l, l − ) · ω ( s − h − l, t, t − l − ) ω − ( f ghs, s − h − , k ) ω − ( s, s − , h − ) · ω − ( t, t − , l − ) ω − ( s − h − lt, t − l − , k ) ψ − ( gh, h − ) ψ − ( h − l, l ) β − ( s, s − ) · β − ( t, t − ) [( δ f ⊗ δ gh ) ⊗ δ s ] ⊗ A [[( δ s − ⊗ δ h − l ) ⊗ δ t ] ⊗ A [( δ t − ⊗ δ l − ) ⊗ δ k ]] . In order to match the basis, let us make the following changes of labelings:- For the α (∆ ⊗ A id)∆ term, take l − = a and h − = b ;- For the (id ⊗ A ∆)∆ term, take h − l = a and l − = b , and swap the role of s and t .With these changes, and with | K | − | L | − , we get that the coefficient of(( δ f ⊗ δ gb − a − ) ⊗ δ t ) ⊗ A [(( δ t − ⊗ δ a ) ⊗ δ s ) ⊗ A (( δ s − ⊗ δ b ) ⊗ δ k )]is, one on hand, equal to ω ( f, gb − , b ) ω ( f, gb − a − , a ) ω ( f gb − a − , t, t − a ) ω ( f gb − , s, s − b ) · ω − ( f gb − a − t, t − as, s − bk ) ω − ( f gb − s, s − b, k ) ω − ( s, s − , b ) ω − ( t, t − , a ) · ω − ( f gb − a − t, t − a, s ) ψ − ( gb − , b ) ψ − ( gb − a − , a ) β − ( s, s − ) β − ( t, t − ) , and, on the other hand, is equal to ω ( f, g ( ab ) − , ab ) ω ( f g ( ab ) − , t, t − ab ) ω ( t − , a, b ) ω ( t − a, s, s − b ) · ω − ( f g ( ab ) − t, t − ab, k ) ω − ( t, t − , ab ) ω − ( s, s − , b ) ω − ( t − as, s − b, k ) · ψ − ( g ( ab ) − , ab ) ψ − ( a, b ) β − ( t, t − ) β − ( s, s − ) . These two are equal by applying (5.6) with θ = ψ and ( f , f , f ) = ( gb − a − , a, b ),and by applying (5.4) with ( g , g , g , g ) = ( t, t − , a, b ), ( f g ( ab ) − , t, t − a, b ),( f, g ( ab ) − , a, b ), ( f g ( ab ) − t, t − as, s − b, k ), ( f g ( ab ) − t, t − a, s, s − b ) . (cid:3) Proposition C.5.
We have that ε satisfies the counit axiom with respect to ∆ .Proof. Take A := A ( K, β ) and B := A ( L, ψ ), and let
ABA := A K,β ( L, ψ ). Since l AABA π AA,ABA = ( m A ⊗ id B ⊗ id A )( α − A,A,B ⊗ id A ) α − A,AB,A , it suffices to show that( m A ⊗ id B ⊗ id A )( α − A,A,B ⊗ id A ) α − A,AB,A ( ε ABA ⊗ id ABA )∆ ′ ABA = id
ABA , for π AABA,ABA ∆ ′ ABA = ∆
ABA in order to establish the left counit axiom. We havethat ( m A ⊗ id B ⊗ id A )( α − A,A,B ⊗ id A ) α − A,AB,A ( ε ABA ⊗ id ABA )∆ ′ ABA [( δ f ⊗ δ g ) ⊗ δ k ]= | K | − | L | − ( m A ⊗ id B ⊗ id A )( α − A,A,B ⊗ id A ) α − A,AB,A ( ε ABA ⊗ id ABA ) (cid:0) L h ∈ L,s ∈ K ω ( f, gh, h − ) ω ( fgh, s, s − h − ) ω − ( fghs, s − h − , k ) ω − ( s, s − , h − ) · ψ − ( gh, h − ) β − ( s, s − ) [(( δ f ⊗ δ gh ) ⊗ δ s ) ⊗ (( δ s − ⊗ δ h − ) ⊗ δ k )] (cid:1) = | K | − | L | − ( m A ⊗ id B ⊗ id A )( α − A,A,B ⊗ id A ) α − A,AB,A (cid:0) L h ∈ L,s ∈ K ω ( f, gh, h − ) ω ( fgh, s, s − h − ) ω − ( fghs, s − h − , k ) ω − ( s, s − , h − ) · ψ − ( gh, h − ) β − ( s, s − ) | L | δ gh,e β ( f, s ) [ δ fs ⊗ (( δ s − ⊗ δ h − ) ⊗ δ k )] (cid:1) = | K | − (cid:0) L h ∈ L,s ∈ K ω ( fs, s − h − , k ) ω ( fs, s − , h − ) ω ( f, gh, h − ) ω ( fgh, s, s − h − ) · ω − ( fghs, s − h − , k ) ω − ( s, s − , h − ) ψ − ( gh, h − ) · β − ( s, s − ) β ( f, s ) β ( fs, s − ) δ gh,e [( δ f ⊗ δ h − ) ⊗ δ k ] (cid:1) = | K | − (cid:0) L s ∈ K ω ( fs, s − g, k ) ω ( fs, s − , g ) ω ( f, s, s − g ) ω − ( fs, s − g, k ) ω − ( s, s − , g ) · β − ( s, s − ) β ( f, s ) β ( fs, s − ) [( δ f ⊗ δ g ) ⊗ δ k ] (cid:1) = ( δ f ⊗ δ g ) ⊗ δ k , where the last equality holds by (5.6) with θ = β and ( f , f , f ) = ( f, s, s − ), andby (5.4) with ( g , g , g , g ) = ( f, s, s − , g ).On the other hand, r AABA π AABA,A = (id A ⊗ id B ⊗ m A ) α AB,A,A . So, it suffices toshow that (id A ⊗ id B ⊗ m A ) α AB,A,A (id
ABA ⊗ ε ABA )∆ ′ ABA = id
ABA , for π AABA,ABA ∆ ′ ABA = ∆
ABA in order to establish the right counit axiom. We havethat (id A ⊗ id B ⊗ m A ) α AB,A,A (id
ABA ⊗ ε ABA )∆ ′ ABA [( δ f ⊗ δ g ) ⊗ δ k ]= | K | − | L | − (id A ⊗ id B ⊗ m A ) α AB,A,A (id
ABA ⊗ ε ABA ) (cid:0) L h ∈ L,s ∈ K ω ( f, gh, h − ) ω ( fgh, s, s − h − ) ω − ( fghs, s − h − , k ) ω − ( s, s − , h − ) · ψ − ( gh, h − ) β − ( s, s − ) [(( δ f ⊗ δ gh ) ⊗ δ s ) ⊗ (( δ s − ⊗ δ h − ) ⊗ δ k )] (cid:1) = | K | − | L | − (id A ⊗ id B ⊗ m A ) α AB,A,A (cid:0) L h ∈ L,s ∈ K ω ( f, gh, h − ) ω ( fgh, s, s − h − ) ω − ( fghs, s − h − , k ) ω − ( s, s − , h − ) · ψ − ( gh, h − ) β − ( s, s − ) | L | δ h − ,e β ( s − , k ) [(( δ f ⊗ δ gh ) ⊗ δ s ) ⊗ δ s − k ] (cid:1) LGEBRAIC STRUCTURES IN GROUP-THEORETICAL FUSION CATEGORIES 53 = | K | − (cid:0) L h ∈ L,s ∈ K ω ( fgh, s, s − k ) ω ( f, gh, h − ) ω ( fgh, s, s − h − ) · ω − ( fghs, s − h − , k ) ω − ( s, s − , h − ) ψ − ( gh, h − ) · β − ( s, s − ) β ( s − , k ) β ( s, s − k ) δ h − ,e [( δ f ⊗ δ gh ) ⊗ δ k ] (cid:1) = | K | − (cid:0) L s ∈ K ω ( fg, s, s − k ) ω ( fg, s, s − ) ω − ( fgs, s − , k ) · β − ( s, s − ) β ( s − , k ) β ( s, s − k ) [( δ f ⊗ δ g ) ⊗ δ k ] (cid:1) = ( δ f ⊗ δ g ) ⊗ δ k , where the last equality holds by (5.6) with θ = β and ( f , f , f ) = ( s, s − , k ), andby (5.4) with ( g , g , g , g ) = ( f g, s, s − , k ). (cid:3) Proposition C.6.
The maps m and ∆ satisfy the Frobenius conditions.Proof. To check the first Frobenius condition, (id ⊗ A m ) α (∆ ⊗ A id) = ∆ m , weapply both sides of the equation to the element (( δ f ⊗ δ g ) ⊗ δ k ) ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ ).On the left hand side we have(id ⊗ A m ) α (∆ ⊗ A id)[(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )]= | K | − | L | − L h ∈ L,s ∈ K δ kf ′ ,e ω ( f, gh, h − ) ω ( f gh, s, s − h − ) · ω − ( f ghs, s − h − , k ) ω − ( s, s − , h − ) ω − ( f ghs, s − h − k, f ′ g ′ k ′ ) · ω ( s − h − k, f ′ g ′ , k ′ ) ω ( k, f ′ , g ′ ) ω − ( s − h − , k, f ′ g ′ ) ω − ( s − , h − , g ′ ) · ψ − ( gh, h − ) ψ ( h − , g ′ ) β ( k, f ′ ) β − ( s, s − ) · [(( δ f ⊗ δ gh ) ⊗ δ s ) ⊗ A (( δ s − ⊗ δ h − g ′ ) ⊗ δ k ′ )] . By applying the change of variables h g ′ h and the substitution f ′ = k − via theterm δ kf ′ ,e , we get the quantity above is equal to | K | − | L | − L h ∈ L,s ∈ K ω ( f, gg ′ h, h − g ′− ) ω ( f gg ′ h, s, s − h − g ′− ) · ω − ( f gg ′ hs, s − h − g ′− , k ) ω − ( s, s − , h − g ′− ) · ω − ( f gg ′ hs, s − h − g ′− k, k − g ′ k ′ ) ω ( s − h − g ′− k, k − g ′ , k ′ ) · ω ( k, k − , g ′ ) ω − ( s − h − g ′− , k, k − g ′ ) ω − ( s − , h − g ′− , g ′ ) · β − ( s, s − ) β ( k, k − ) ψ − ( gg ′ h, h − g ′− ) ψ ( h − g ′− , g ′ ) · [(( δ f ⊗ δ gg ′ h ) ⊗ δ s ) ⊗ A (( δ s − ⊗ δ h − ) ⊗ δ k ′ )] . On the other hand, we get∆ m [(( δ f ⊗ δ g ) ⊗ δ k ) ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ )](C.7) = | K | − | L | − L h ∈ L,s ∈ K ω ( f gk, k − g ′ , k ′ ) ω ( k, k − , g ′ ) ω − ( f g, k, k − g ′ ) · ω − ( f, g, g ′ ) ω ( f, gg ′ h, h − ) ω ( f gg ′ h, s, s − h − ) ω − ( f gg ′ hs, s − h − , k ′ ) · ω − ( s, s − , h − ) ψ ( g, g ′ ) ψ − ( gg ′ h, h − ) β ( k, k − ) β − ( s, s − ) · [(( δ f ⊗ δ gg ′ h ) ⊗ δ s ) ⊗ A (( δ s − ⊗ δ h − ) ⊗ δ k ′ )] . where we have made the same substitution f ′ = k − . After cancelling the terms ω ( k, k − , g ′ ), β ( k, k − ) and β − ( s, s − ), the coefficients of the two sides of the equa-tion are equal due to applying (5.6) with θ = ψ and ( f , f , f ) = ( gg ′ h, h − g ′− , g ′ ),and applying (5.4) with ( g , g , g , g ) = ( f gg ′ hs, s − h − g ′− , k, k − g ′ ),( f gg ′ hs, s − h − g ′− k, k − g ′ , k ′ ), ( f gg ′ h, s, s − h − g ′− , g ′ ), ( f, gg ′ h, h − g ′− , g ′ ),( s, s − , h − g ′− , g ′ ).We check the second Frobenius property, ∆ m = ( m ⊗ A id) α − (id ⊗ A ∆). Byapplying both sides of the equation to (( δ f ⊗ δ g ) ⊗ δ k ) ⊗ A (( δ f ′ ⊗ δ g ′ ) ⊗ δ k ′ ), weget the left hand side coefficients of ( δ f ⊗ δ gg ′ h ⊗ δ s ) ⊗ A (( δ s − ⊗ δ h − ) ⊗ δ k ′ ) givenby (C.7). On the right hand side, the coefficient is ω ( k − , g ′ h, h − ) ω ( k − g ′ h, s, s − h − ) ω − ( k − g ′ hs, s − h − , k ′ ) · ω − ( s, s − , h − ) ω ( f gk, k − g ′ hs, s − h − k ′ ) ω ( f gk, k − g ′ h, s ) · ω ( k, k − , g ′ h ) ω − ( f g, k, k − g ′ h ) ω − ( f, g, g ′ h ) · ψ − ( g ′ h, h − ) ψ ( g, g ′ h ) β − ( s, s − ) β ( k, k − ) . After cancelling the terms ω − ( s, s − , h − ), β ( k, k − ) and β − ( s, s − ), both coeffi-cients are equal due to applying (5.6) with θ = ψ and ( f , f , f ) = ( g, g ′ h, h − ) andthen applying (5.4) with the values of ( g , g , g , g ) assigned sequentially as follows:( k, k − , g ′ h, h − ); ( f, g, g ′ h, h − ); ( f g, k, k − g ′ h, h − ); ( f gk, k − g ′ hs, s − h − , k ′ );( f gk, k − g ′ h, s, s − h − )). (cid:3) References [1] B. Bakalov and A. Kirillov, Jr.
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Morales: Departamento de Matem´aticas, Universidad de los Andes, Cra. 1
E-mail address : [email protected] M¨uller: Departamento de Matem´atica e Estat´ıstica, Universidade Federal de S˜aoJo˜ao del-Rei, Prac¸a Frei Orlando, 170, Centro, S˜ao Jo˜ao del-Rei, Minas Gerais, Brazil,CEP: 36307-352
E-mail address : [email protected] Ros Camacho: School of Mathematics, Cardiff University, Senghennydd Road, CF244AG Cardiff, Wales
E-mail address : [email protected] Plavnik: Department of Mathematics, Indiana University, Bloomington, IN 47405USA
E-mail address : [email protected] Tabiri: African Institute for Mathematical Sciences Ghana, Summerhill Estates,GPS: GK- 0647-1372, Accra, Ghana
E-mail address : [email protected] Walton: Department of Mathematics, University of Illinois at Urbana-Champaign,Urbana, IL 61801 USA
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