On the notion of exact sequence: from Hopf algebras to tensor categories
aa r X i v : . [ m a t h . QA ] M a r ON THE NOTION OF EXACT SEQUENCE: FROM HOPFALGEBRAS TO TENSOR CATEGORIES
SONIA NATALE
Abstract.
We present an overview of the notions of exact sequences of Hopfalgebras and tensor categories and their connections. We also present someexamples illustrating their main features; these include simple fusion categoriesand a natural question regarding composition series of finite tensor categories.
Contents
1. Introduction 22. Preliminaries on Hopf algebras and tensor categories 33. Exact sequences of Hopf algebras 43.1. Normality of Hopf subalgebras and Hopf algebra maps 53.2. Example: Abelian exact sequences and matched pairs of groups 63.3. Jordan-H¨older theorem for finite dimensional Hopf algebras 73.4. Semisolvability and related questions 73.5. Simple Hopf algebras 84. Exact sequences of tensor categories 104.1. Examples from exact sequences of Hopf algebras 104.2. Extensions and normal Hopf monads 124.3. Perfect exact sequences and central commutative algebras 124.4. Examples from finite groups: equivariantization and crossed extensionsby matched pairs 134.5. Abelian exact sequences of tensor categories and matched pairs ofgroups 165. Exact sequences of finite tensor categories with respect to a modulecategory 165.1. Exact sequences and exact factorizations 185.2. Exact factorizations arising from exact sequences of finite tensorcategories. 196. Examples and open questions 196.1. Fusion subcategories of index 2 need not be normal 206.2. Further examples of simple fusion categories of dimension p a q b Date : March 30, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Hopf algebra; tensor category; fusion category; exact sequence.Partially supported by CONICET and SeCYT–UNC. Introduction
Perhaps the most natural way to understand or attempt a classification of agiven algebraic structure is by decomposing it into ’simpler’ structures. A funda-mental example of this principle is provided by the theory of finite groups. Everyfinite simple group G has a composition series whose factors are finite simple groupsuniquely determined by G , up to permutations. In particular, G can be built upby means of a number of successive extensions of finite simple groups. In this con-text, both the classification of the simple structures -finite simple groups- and theclassification of the possible extensions -governed by suitable cohomology theories-are hard questions that involve deep mathematical techniques.In this paper we aim to present an overview of an analogous approach to thestudy of structures that generalize that of groups, namely, Hopf algebras and tensorcategories. We discuss the notion of extension in each of these contexts and someof their main features. More precisely, we focus on extensions arising from exactsequences: these do not include certain extensions arising from group gradings ontensor categories which play an important role in the classification of certain classesof fusion categories [18].Very little is known about a possible approach to classify simple structures inthese cases. We discuss several simple examples that show some contrast with thetheory of finite groups.We start by recalling the notion of exact sequence of Hopf algebras in Section 3.The main contributions towards this notion appeared in the work of G. I. Kac [25],M. Takeuchi [57], W. Singer [54], B. Parshall and J. P. Wang [50], H.-J. Schneider[53], S. Majid [28], N. Andruskiewitsch and J. Devoto [5], [1], Hofstetter [23] andothers.Exact sequences of Hopf algebras generalize exact sequences of groups. In asimilar vein, the notion of normal subgroup and simple group have generalizationsto the notions of normal Hopf subalgebra and simple Hopf algebra, that we discussin Subsections 3.1 and 3.5.As in the case of finite groups, every finite dimensional Hopf algebra H has a composition series : these are sequences of simple Hopf algebras H , . . . , H n , calledthe factors of the series, defined as follows [6]: If H is simple, then n = 1 and H = H ; if on the other hand, H contains a proper normal Hopf subalgebra A ,and A , . . . , A m , B , . . . , B ℓ , are composition series of A and of the quotient Hopfalgebra B = H/HA + , respectively, then n = m + ℓ and H i = A i , if 1 ≤ i ≤ m, H i = B i − m , if m < i ≤ m + ℓ. Furthermore, a Jordan-H¨older theorem holds in the context of finite dimensionalHopf algebras [42]. This and related facts are discussed in Subsections 3.3 and 3.4.In Section 4.1 we review the notion of exact sequence of tensor categories de-veloped with A. Brugui`eres in [8], [9] and its relation with Hopf monads and com-mutative central algebras. We also discuss in this section a family of examplesarising from so-called crossed actions of a matched pair of finite groups [43] andrecent classification results for extensions of a fusion category by the category ofrepresentations of a finite group obtained in [47].The notion of exact sequence of tensor categories discussed in Section 4.1 wasextended to the notion of exact sequence with respect to a module category byP. Etingof and S. Gelaki [14]: In this sense, an exact sequence of (finite) tensorcategories becomes an exact sequence with respect to a rank-one module category.
XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 3
We overview the definition of [14] in Section 5 as well as its connection with exactfactorizations of tensor categories from [20], [34], [51].We end the paper by presenting some examples that answer a number of naturalquestions regarding the behaviour of exact sequences of tensor categories. This isdone in Section 6. In contrast with the properties enjoyed by groups and Hopfalgebras, these examples show in particular that: • Fusion subcategories of index 2 need not be normal with respect to a modulecategory (Subsection 6.1); • There exist fusion categories of Frobenius-Perron dimension p a q b , where p and q are prime numbers, which are simple with respect to any modulecategory (Subsection 6.2).Regarding the question of formulating a Jordan-H¨older theorem for finite tensorcategories (see [46, Question 4.7]), we consider a natural generalization of the notionof composition series of finite dimensional Hopf algebra to the context of finitetensor categories: namely, we call a sequence of finite tensor categories C , . . . , C n -the factors of the series- a composition series of a finite tensor category C if, asbefore, n = 1 and C = C if C is does not fit into any exact sequence with respect toa module category, while if C ′ −→ C −→ C ′′ ⊠ End( M ) is an exact sequence withrespect to some C ′ -module category M such that FPdim C ′ , FPdim C ′′ >
1, then n = m + ℓ and C i = C ′ i , 1 ≤ i ≤ m , C i = C ′′ i − m , m < i ≤ m + ℓ , where C ′ , . . . , C ′ m , C ′′ , . . . , C ′′ ℓ , are composition series of C ′ and C ′′ , respectively.One of the main new contributions of this paper is a negative answer to thefollowing question: Question 1.1.
Is it true that two composition series of a finite tensor categorythus defined have the same factors up to a permutation?
Indeed we show that the answer to Question 1.1 is negative even in the contextof (braided non-degenerate) fusion categories. Hence composition series of fusioncategories thus defined fail to satisfy a Jordan-H¨older theorem. This is done inSubsection 3.3; see Corollary 6.8 and Remark 6.9.In spite of this fact, we mention that an analogue of the Jordan-H¨older theoremdoes hold for weakly group-theoretical fusion categories introduced in [18]. Thedefinition of a composition series for this kind of category is given in terms of groupequivariantizations and group graded extensions and the composition factors, whichare Morita invariants, are finite simple groups [44].Finally, we include in Subsection 6.6 some questions that we believe are inter-esting in relation with the notions discussed previously.2.
Preliminaries on Hopf algebras and tensor categories
We shall work over an algebraically closed field k of characteristic zero. A tensorcategory over k is a k -linear abelian category with finite dimensional Hom spacesand objects of finite length, endowed with a rigid monoidal category structure, suchthat the monoidal product is k -linear in each variable and the unit object is simple.A tensor category over k is called finite if it is equivalent as a k -linear category tothe category of finite dimensional left modules over a finite dimensional k -algebra.A fusion category over k is a semisimple finite tensor category. We refer the readerto [16] for a systematic study of tensor categories.Let C and D be tensor categories over k . A tensor functor F : C → D is a k -linear exact (strong) monoidal functor F . SONIA NATALE
Let G be a group. We shall denote by Vec G the tensor category of finite dimen-sional G -graded vector spaces and by Rep G the tensor category of finite dimensional k -linear representations of G .Let H be a Hopf algebra over k . We shall indicate by H + the augmentationideal of H , defined as H + = { x ∈ H : ε ( x ) = 0 } . Except for Section 3, all Hopfalgebras we consider in this paper are assumed to have a bijective antipode (this isautomatically true if the Hopf algebra is finite dimensional). Thus the categories H -mod of finite dimensional (left) H -modules and comod- H of finite dimensional(right) H -comodules are tensor categories over k .Let C be a tensor category. A left module category over C , or C -module category ,is a k -linear abelian category M together with a k -linear bi-exact functor ⊗ : C × M → M , and natural isomorphisms m X,Y,M : ( X ⊗ Y ) ⊗ M → X ⊗ ( Y ⊗ M ), u M : ⊗ M → M , X, Y ∈ C , M ∈ M , satisfying natural associativity and unitary conditions.Let M and N be C -module categories. A C - module functor M → N is a k -linearfunctor F : M → N endowed with natural isomorphisms F ( X ⊗ M ) → X ⊗ F ( M ) , for all X ∈ C , M ∈ M , satisfying appropriate conditions. A C -module functor iscalled an equivalence of C -module categories if it is an equivalence of categories. Amodule category M is called indecomposable if it is not equivalent as a C -modulecategory to the direct sum of two nontrivial module categories.Suppose C is a finite tensor category. A finite left C -module category M is exact if for every projective object P ∈ C and for every M ∈ M , P ⊗ M is aprojective object of M . Let M be an indecomposable exact C -module category.Then the category End C ( M ) of right exact C -module endofunctors of M is a finitetensor category. A tensor category D is (categorically) Morita equivalent to C if D ∼ = End C ( M ) op for some exact indecomposable C -module category M .3. Exact sequences of Hopf algebras
Let H be a Hopf algebra over k . The aim of this section is to give an account ofsome of the main features regarding the notion of exact sequences of Hopf algebrasand present some examples.Let π : H → B be a Hopf algebra map. The subalgebras coπ H and H coπ of leftand right B -coinvariants of H are defined, respectively, by coπ H = { h ∈ H : ( π ⊗ id)∆( h ) = 1 ⊗ h } ,H coπ = { h ∈ H : (id ⊗ π )∆( h ) = h ⊗ } . Definition 3.1. ([5].) An exact sequence of Hopf algebras is a sequence of Hopfalgebra maps(3.1) k −→ H ′ i −→ H π −→ H ′′ −→ k, satisfying the following conditions:(a) i is injective and π is surjective,(b) ker π = Hi ( H ′ ) + ,(c) i ( H ′ ) = coπ H .A Hopf algebra H fitting into an exact sequence (3.1) is called an extension of H ′′ by H ′ . XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 5
Note that either of the conditions (b) or (c) in Definition 3.1 implies that πi = ǫ H ′ → H ′′ . If H is faithfully flat over H ′ , then (a) and (b) imply (c). Dually, if H is faithfully coflat over H ′′ , then (a) and (c) imply (b).We refer the reader to [5], [53], [57] for further details on the notion of an exactsequence.Let (3.1) be an exact sequence of finite dimensional Hopf algebras. Then, asa consequence of the Nichols-Zoeller freeness theorem, it is cleft , that is, the map π : H → B admits a convolution invertible B -colinear and A -linear section B → H .This implies that H isomorphic as a Hopf algebra to a bicrossed product A B withrespect to suitable compatible data. See [ ? ], [5]. In particulardim H = dim H ′ dim H ′′ . In addition, H is semisimple if and only if H ′ and H ′′ are semisimple. Definition 3.2.
We shall say that a Hopf algebra H is simple if it does not fit intoany exact sequence of Hopf algebras (3) with H ′ ≇ k and H ′′ ≇ k .3.1. Normality of Hopf subalgebras and Hopf algebra maps.
Consider theleft and right adjoint actions of H on itself defined, respectively, by(3.2) h.a = h (1) a S ( h (2) ) , a.h = S ( h (1) ) ah (2) , for all a, h ∈ H .Formulas (3.2) generalize those defining the adjoint actions of a group. In thisvein, the notion of a normal subgroup can be generalized to that of a normal Hopfsubalgebra; that is, a Hopf subalgebra K of H is called normal if it is stable underboth action actions (3.2).Dually, the left and right adjoint coactions of H are defined, respectively, by ρ ℓ : H → H ⊗ H, ρ ℓ ( h ) = h (1) S ( h (3) ) ⊗ h (2) ,ρ r : H → H ⊗ H, ρ r ( h ) = h (2) ⊗ h (1) S ( h (3) ) . Let π : H → H ′′ be a Hopf algebra map. The map π is called normal if thekernel I of π is a subcomodule for both adjoint coactions of H .Let H ′ ⊆ H be a normal Hopf subalgebra. Then H ( H ′ ) + = ( H ′ ) + H is a Hopfideal of H and the canonical map H → H/H ( H ′ ) + is a Hopf algebra map. If H isfaithfully flat over H ′ , then there is an exact sequence of Hopf algebras k −→ H ′ −→ H −→ H/H ( H ′ ) + −→ k. Similarly, if π : H → H ′′ is a surjective normal Hopf algebra map, then coπ H = H coπ is a Hopf subalgebra and if H is faithfully coflat over H ′′ , there is an exactsequence of Hopf algebras k −→ coπ H −→ H π −→ H ′′ −→ k. Suppose H is a finite dimensional Hopf algebra. Then H is simple if and onlyif H contains no proper normal Hopf subalgebra if and only if it admits no propernormal quotient Hopf algebra.Furthermore, a sequence of Hopf algebra maps k −→ H ′ i −→ H π −→ H ′′ −→ k is an exact sequence if and only if the dual sequence k −→ ( H ′′ ) ∗ π ∗ −→ H ∗ i ∗ −→ ( H ′ ) ∗ −→ k is exact. Therefore the notion of simplicity of a finite dimensional Hopf algebra isself-dual, that is, H is simple if and only if H ∗ is simple. SONIA NATALE
Example: Abelian exact sequences and matched pairs of groups.
Acelebrated source of examples of Hopf algebra extensions arises from matched pairsof groups: these are called abelian extensions and were introduced in the early workof G. I. Kac, W. Singer, S. Majid and M. Takeuchi [25], [54], [28], [56]. We referthe reader to [31], [32] for a detailed study of the cohomology theory underlying anabelian exact sequence.
Definition 3.3.
An exact sequence of finite dimensional Hopf algebras(3.3) k −→ H ′ i −→ H π −→ H ′′ −→ k is called an abelian exact sequence if H ′ is commutative and H ′′ is cocommutative.Since our base field k is algebraically closed of characteristic zero, this meansthat H ′ ∼ = k Γ and H ′′ ∼ = kG , for some finite groups Γ , G .The exactness of the sequence (3.3) allows to endow the pair ( G, Γ) with thestructure of a matched pair of groups. That is, (3.3) gives rise to actions by per-mutations Γ ⊳ ←− Γ × G ⊲ −→ G such that(3.4) s ⊲ xy = ( s ⊲ x )(( s ⊳ x ) ⊲ y ) , st ⊳ x = ( s ⊳ ( t ⊲ x ))( t ⊳ x ) , for all s, t ∈ Γ, x, y ∈ G .Given groups G and Γ, the data of a pair of compatible actions making ( G, Γ)into a matched pair of groups is equivalent to the data of a group E together withan exact factorization into subgroups (isomorphic to) G and Γ: that is, a group E such that E = G Γ and G ∩ Γ = { e } . In such situation, the relevant actions ⊲ and ⊳ are determined by the relations gx = ( g ⊲ x )( g ⊳ x ) , for every x ∈ G , g ∈ Γ.Fix a matched pair of finite groups ( G, Γ). Consider the left action of G on k Γ defined by ( x.f )( g ) = f ( g ⊳ x ), f ∈ k Γ , and let σ : G × G → ( k ∗ ) Γ bea normalized 2-cocycle. Dually, consider the right action of Γ on k G given by( w.g )( x ) = w ( x ⊲ g ), w ∈ k G , and let τ : Γ × Γ → ( k ∗ ) G be a normalized 2-cocycle.Under appropriate compatibility conditions between σ and τ , the vector space H = k Γ ⊗ kG becomes a (semisimple) Hopf algebra, denoted H = k Γ τ σ kG ,with the crossed product algebra structure and the crossed coproduct coalgebrastructure; see [31],[32]. For all g, h ∈ Γ, x, y ∈ G , we have( e g x )( e h y ) = δ g ⊳ x,h σ g ( x, y ) e g xy, (3.5) ∆( e g x ) = X st = g τ x ( s, t ) e s t ⊲ x ) ⊗ e t x, (3.6)where σ s ( x, y ) = σ ( x, y )( s ) and τ x ( s, t ) = τ ( s, t )( x ), s, t ∈ Γ, x, y ∈ G .Let i : k Γ → H = k Γ τ σ kG and π : H = k Γ τ σ kG → kF be the Hopf algebramaps defined by i ( f ) = f e , π ( f g ) = ε ( f ) g , f ∈ k Γ , g ∈ G . The Hopf algebra H fits into an abelian exact sequence k −→ k Γ i −→ H π −→ kG −→ k. Moreover, every Hopf algebra H fitting into an abelian exact sequence (3.3) isisomorphic to a bicrossed product k Γ τ σ kF for an appropriate matched pair ( G, Γ)and compatible actions and cocycles σ and τ . Equivalence classes of such extensionsassociated to a fixed matched pair ( G, Γ) form an abelian group Opext( k Γ , kG ),whose unit element is the class of the split extension k Γ kG . XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 7
An example of an abelian extensions is given by the Drinfeld double D ( G ) of afinite group G : the Hopf algebra D ( G ) fits into an exact sequence k −→ k G −→ D ( G ) −→ kG −→ k. In the associated matched pair (
G, G ), ⊳ : G × G → G is the adjoint action of G onitself, while ⊲ : G × G → G is the trivial action.3.3. Jordan-H¨older theorem for finite dimensional Hopf algebras.
Thequestion of establishing an analogue of the Jordan-H¨older theorem of group the-ory for finite dimensional Hopf algebras was raised by N. Andruskiewitsch in [2,Question 2.1] and answered in [42].We start by recalling the followig definition given in [6].
Definition 3.4.
Let H be a finite dimensional Hopf algebra over a field k . A composition series of H is a sequence of finite dimensional simple Hopf algebras H , . . . , H n defined recursively as follows: If H is simple, we let n = 1 and H = H .If k ( A ( H is a normal Hopf subalgebra, and A , . . . , A m , B , . . . , B ℓ , arecomposition series of A and B = H/HA + , respectively, then we let n = m + ℓ and H i = A i , if 1 ≤ i ≤ m, H i = B i − m , if m < i ≤ m + ℓ. The Hopf algebras H , . . . , H n are called the factors of the series. The number n iscalled the length of the series.Every finite dimensional Hopf algebra admits a composition series. The followingis an analogue of the Jordan-H¨older theorem for finite dimensional Hopf algebras: Theorem 3.5. ([42, Theorem 1.2].)
Let H , . . . , H n and H ′ , . . . , H ′ m be two com-position series of H . Then there exists a bijection f : { , . . . , n } → { , . . . , m } suchthat H i ∼ = H ′ f ( i ) as Hopf algebras. Let H be a finite dimensional Hopf algebra and let H , . . . , H n be a compositionseries of H . The simple Hopf algebras H i , 1 ≤ i ≤ n , are called the compositionfactors of H . The number n is called the length of H .Let us consider for instance the case of abelian extensions. Let G and Γ be finitegroups and let H be an abelian extension of k Γ by kG . Then the compositionfactors of H are the group algebras of the composition factors of G and the dualgroup algebras of the composition factors of Γ. See [42, Example 4.7].In particular, if G , . . . , G n are the composition factors of the finite group G ,then the composition factors of its Drinfeld double D ( G ) are the Hopf algebras k G , . . . , k G n , kG , . . . , kG n .3.4. Semisolvability and related questions. A lower subnormal series of aHopf algebra H is a series of Hopf subalgebras(3.7) k = H n ⊆ H n − ⊆ · · · ⊆ H ⊆ H = H, such that H i +1 is a normal Hopf subalgebra of H i , for all i . The factors of theseries (3.7) are the quotient Hopf algebras H i /H i H + i +1 , i = 0 , . . . , n − upper subnormal series of H is a series of surjective Hopf algebramaps(3.8) H = H (0) → H (1) → · · · → H ( n ) = k, such that H ( i +1) is a normal quotient Hopf algebra of H ( i ) , for all i = 0 , . . . , n − factors of (3.8) are the Hopf algebras co H ( i +1) H ( i ) ⊆ H ( i ) , i = 0 , . . . , n − SONIA NATALE A lower (respectively upper ) composition series of H is a lower (respectively,upper) subnormal series which does not admit any proper refinement. See [42,Section 5].A Hopf algebra H is called lower-semisolvable (respectively, upper-semisolva-ble ) if it admits a lower (respectively, upper) subnormal series whose factors arecommutative or cocommutative [36]. We shall say that H is semisolvable if it iseither lower or upper semisolvable.Every semisolvable finite dimensional Hopf algebra is semisimple and cosemisim-ple. On the other hand, every semisimple Hopf algebra of dimension p n , p a primenumber, is semisolvable [30], [36]. Moreover, if the group G is nilpotent then anytwisting of the group Hopf algebra kG is semisolvable [19].In [42] we proved analogues of the Zassenhaus’ butterfly lemma and the Schreier’srefinement theorem for finite dimensional Hopf algebras and, following the lines ofthe classical proof in group theory, applied them to prove an analogue of the Jordan-H¨older theorem for lower and upper composition series of H .Thus the lower and upper composition factors of H and its lower and upperlengths, which are also well-defined invariants of H , were introduced. In contrastwith the case of the composition factors, the lower or upper composition factors arenot necessarily simple as Hopf algebras. This motivates the question of deciding ifthere is an intrinsic characterization of the Hopf algebras that can arise as lowercomposition factors [42, Question 5.7].Some properties of lower and upper composition factors and their relation withthe composition factors were studied in [42]. Unlike for the case of the length, thelower and upper lengths are not additive with respect to exact sequences and theyare not invariant under duality in general.Neither the composition factors nor the upper or lower composition factors of afinite dimensional Hopf algebra H are categorical invariants of H . In other words,they are not invariant under twisting deformations of H . In fact, there existsa (semisimple) Hopf algebra H such that H is simple as a Hopf algebra and H is twist equivalent to the group algebra of a solvable group G (see Theorem 3.6below). In particular, the categories of finite dimensional representations of H and G are equivalent fusion categories.3.5. Simple Hopf algebras.
Recall that a finite dimensional Hopf algebra is sim-ple if it contains no proper normal Hopf subalgebras. For instance, if G is a finitesimple group, then the group algebra kG and its dual k G are simple Hopf algebras.Furthermore, in this case, any twisting deformation of kG is simple [48]. However,there are examples of solvable groups that admit simple twisting deformations.Finite dimensional Hopf algebras with tensor equivalent categories of represen-tations are obtained from one another by a twisting deformation. Properties of H invariant under twisting are of special interest because they depend only on thetensor category H -mod.In the paper [19] we presented examples showing that the notions of simplicityand (semi)solvability of a Hopf algebra are not twist invariants; that is, they arenot categorical notions.Let p , r and q be prime numbers such that q divides p − r −
1. There isa family of supersolvable groups G of order prq that can be deformed through atwist into nontrivial simple Hopf algebras. XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 9
Let us recall this construction. Let G = Z p ⋊ Z q and G = Z r ⋊ Z q be theonly nonabelian groups of orders pq and rq , respectively. Let G = G × G and let S ∼ = Z q × Z q be a subgroup of G order q . In particular, G is supersolvable and Z ( G ) = 1.Let 1 = ω ∈ H ( b S, k ∗ ), J ∈ kG ⊗ kG the twist lifted from S corresponding to ω . Let also H = ( kG ) J . Note that the cocycle ω is nondegenerate. Also, H is anoncommutative noncocommutative Hopf algebra of dimension prq . Theorem 3.6. ([19, Theorem 4.5].)
The Hopf algebra H is simple. The proof relied on the comparison of the (co)representation theory of the giventwistings [13] with that of an extension [36].Certain twists of the symmetric group S n on n letters, n ≥
5, were also shownto be simple as Hopf algebras in [19].As a consequence of Theorem 3.6 the analogue of Burnside’s p a q b -Theorem forfinite groups does not hold for semisimple Hopf algebras.Theorem 3.6 provides the smallest example of a noncommutative noncocommu-tative semisimple Hopf algebra which is simple: this appears in dimension 36 as atwisting of D × D . This turns out to be the only simple example in dimension < Theorem 3.7. ([40, Theorem 1].)
Every semisimple Hopf algebra of dimensionless than 60 is semisolvable up to a cocycle twist.
The previously mentioned results on simplicity of twisting deformations provideus with three examples of non-commutative non-cocommutative semisimple Hopfalgebras of dimension 60 which are simple as Hopf algebras. The first two arethe Hopf algebras A and A ≃ A ∗ constructed by D. Nikshych [48]. We have A = ( k A ) J , where J ∈ k A ⊗ k A is an invertible twist lifted from a nondegenerate2-cocycle in a subgroup of A isomorphic to Z × Z .The third example is the self-dual Hopf algebra B constructed in [19]. In this case B = ( kD ⊗ kD ) J , where J is an invertible twist also lifted from a nondegenerate2-cocycle in a subgroup of D × D isomorphic to Z × Z .As coalgebras, these examples are isomorphic to direct sums of full matric coal-gebras, as follows: A ≃ k ⊕ M ( k ) (2) ⊕ M ( k ) ⊕ M ( k ) , (3.9) A ≃ k (12) ⊕ M ( k ) (3) , (3.10) B ≃ k (4) ⊕ M ( k ) (6) ⊕ M ( k ) (2) . (3.11)As for the group-like elements, we have G ( A ) ≃ A and G ( B ) ≃ Z × Z .It was shown in [18, Theorem 9.12] that Rep A ≃ Rep A is the only fusioncategory of dimension 60 which contains no proper fusion subcategories.The following theorem was shown in [41], answering Question 2.4 in [2]. Theorem 3.8. ([41, Theorem 1.4].)
Let H be a nontrivial semisimple Hopf algebraof dimension . Suppose H is simple. Then H is isomorphic to A or to A orto B . Exact sequences of tensor categories
Let C and C ′′ be tensor categories over k . A tensor functor F : C → C ′′ is called dominant (or surjective ) if every object of C ′′ is a subobject of F ( X ) for some object X of C . If C ′′ is a finite tensor category, then F is dominant if and only if everyobject of C ′′ is a subquotient of F ( X ) for some object X of C ; see [14, Lemma 2.3].A tensor functor F : C → C ′′ is called normal if for every object X of C , thereexists a subobject X ⊂ X such that F ( X ) is the largest trivial subobject of F ( X ).For a tensor functor F : C → C ′′ , let Ker F denote the tensor subcategory F − ( h i ) ⊆ C of objects X of C such that F ( X ) is a trivial object of C ′′ . Ifthe functor F has a right adjoint R , then F is normal if and only if R ( ) belongsto Ker F [8, Proposition 3.5]. Definition 4.1.
Let C ′ , C , C ′′ be tensor categories over k . An exact sequence oftensor categories is a sequence of tensor functors(4.1) C ′ f −→ C F −→ C ′′ , such that the tensor functor F is dominant and normal and the tensor functor f isa full embedding whose essential image is Ker F [8].If C fits into an exact sequence (4.1), we say that C is an extension of C ′′ by C ′ .Every exact sequence of tensor categories (4.1) defines a fiber functor on thekernel C ′ : ω = Hom( , F f ) : C ′ → Vec . The induced Hopf algebra H of the exact sequence (4.1) is defined as(4.2) H = coend( ω );see [8, Subsection 3.3]. Thus there is an equivalence of tensor categories C ′ ∼ =comod − H such that the following diagram of tensor functors is commutative: C ′ ∼ = / / ω $ $ ❏❏❏❏❏❏❏❏❏❏ comod − H U (cid:15) (cid:15) Vec . The induced Hopf algebra H of (4.1) is finite-dimensional if and only if the tensorfunctor F admits adjoints [8, Proposition 2.15]. Hence if C ′ and C ′′ are finite tensorcategories, then so is C .The fiber functor ω corresponds to a (rank one) C ′ -module category structureon Vec, that we shall denote by M (see [49, Proposition 4.1]). Assume that H isfinite dimensional. As a consequence of [49, Theorem 4.2] we obtain the followingrelation between M and the induced Hopf algebra:( C ′ ) ∗M ∼ = H -mod . Examples from exact sequences of Hopf algebras.
All Hopf algebrasconsidered in this subsection are assumed to have a bijective antipode.Let f : H → H be a Hopf algebra map. Then f induces by restriction tensorfunctors f ∗ : comod- H → comod- H , f ∗ : H -mod → H -mod . The functor f ∗ : comod- H → comod- H is dominant if and only if the functor (cid:3) H H : Comod- H → Comod- H is faithful. On the other hand, if f is injectiveand H is finite-dimensional, then f ∗ : H -mod → H -mod is dominant. See [8,Lemma 2.11]. XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 11
Recall that the map f has a kernel and a cokernel in the category of Hopfalgebras, called the categorical kernel and the categorical cokernel of f , and defined,respectively, in the formHker( f ) = { h ∈ H | h (1) ⊗ f ( h (2) ) ⊗ h (3) = h (1) ⊗ ⊗ h (2) } Hcoker( f ) = H /H f ( H +1 ) H . The following proposition describes the kernels of the tensor functors induced bya Hopf algebra map.
Proposition 4.2. ([8, Lemma 2.10].)
Let f : H → H ′ be a morphism of Hopfalgebras over a field, and let f ∗ : comod - H → comod - H ′ , f ∗ : H ′ - mod → H - mod be the tensor functors induced by f . Then the following hold:(1) Ker f ∗ = comod - Hker( f ) . Moreover the tensor functor f ∗ is normal if andonly if H coH ′ = coH ′ H , and in this case Hker( f ) = H coH ′ .(2) Ker f ∗ = Hcoker( f ) - mod . Moreover the tensor functor f ∗ is normal if f ( H ) is a normal Hopf subalgebra of H ′ , and in this case Hcoker( f ) = H ′ /H ′ f ( H + ) . We shall say that an exact sequence of Hopf algebras(4.3) k −→ H ′ i −→ H π −→ H ′′ −→ k, is strictly exact if H is faithfully coflat over H ′′ . Remark . Notice that, since the antipode of H is bijective, H is right faithfullycoflat over H ′′ if and only if it is left faithfully coflat over H ′′ . Moreover, if this isthe case, then π is normal if and only if coπ H = H coπ . See [57].Observe in addition that the antipode of H induces an anti-isomorphism ofalgebras S : coπ H → H coπ with inverse S − : H coπ → coπ H . In particular, if i ( H ′ ) = coπ H , then coπ H (being a Hopf subalgebra) is stable under the antipodeand therefore coπ H = H coπ .Assume that the sequence (4.3) is strictly exact. Then, as can be seen from thefacts recalled in Remark 4.3, it is a strictly exact sequence of Hopf algebras in thesense of [53], that is, the following condiditions hold:(a) π is a normal Hopf algebra map,(b) H is right faithfully coflat over H ′′ ,(c) i ( H ′ ) = Hker π .These conditions are furthermore equivalent to the following:(a’) H ′ is a normal Hopf subalgebra of H ,(b’) H is right faithfully flat over H ′ ,(c’) H ′′ = Hcoker i .In this way we obtain: Theorem 4.4. ([8, Proposition 3.9].)
Every strictly exact sequence of Hopf algebras (4.3) gives rise to an exact sequence of tensor categories comod - H ′ i ∗ −→ comod - H π ∗ −→ comod - H ′′ . If in addition H is finite-dimensional, it also gives rise to an exact sequence oftensor categories: H ′′ - mod π ∗ −→ H - mod i ∗ −→ H ′ - mod . For instance, an exact sequence of groups 1 −→ G ′ −→ G −→ G ′′ −→ G ′ −→ Vec G −→ Vec G ′′ , and if G is finite, to an exact sequenceRep G ′′ −→ Rep G −→ Rep G ′ . Extensions and normal Hopf monads.
We refer the reader to [10], [7] forthe notion of Hopf monad on a monoidal category.Let F : C → C ′′ be a tensor functor between tensor categories and assume that F admits a left adjoint G . Then the composition F G : C ′′ → C ′′ is a k -linearright exact Hopf monad on C ′′ , and C is tensor equivalent to the category ( C ′′ ) T of T -modules in C ′′ .The functor F is dominant if and only if T is faithful [8, Proposition 4.1].A Hopf monad T on a tensor category C ′′ is called normal if T ( ) is a trivialobject of C ′′ . When T is the Hopf monad corresponding to F as above, the normalityof T is equivalent to the normality of F [8, Proposition 4.6].Normal faithful Hopf monads classify extensions of tensor categories in view ofthe following theorem: Theorem 4.5. ([8, Theorem 4.8].)
Let C ′ , C ′′ be tensor categories and assume that C ′ is finite. Then the following data are equivalent:(1) A normal faithful k -linear right exact Hopf monad T on C ′′ , with inducedHopf algebra H , endowed with a tensor equivalence K : C ′ ∼ = comod - H ;(2) An extension C ′ → C → C ′′ of C ′′ by C ′ . Under the correspondence established by Theorem 4.5, the induced Hopf algebraof the exact sequence C ′ → C → C ′′ is identified with the Hopf monad of therestriction of T to the trivial subcategory of C ′′ .4.3. Perfect exact sequences and central commutative algebras.
Let C and C ′′ be tensor categories. A tensor functor F : C → C ′′ is called perfect if it admitsan exact right adjoint [9, Subsection 2.1].Every tensor functor F : C → C ′′ between finite tensor categories C and C ′′ isperfect [47, Lemma 2.1].Let F : C → C ′′ be a dominant perfect tensor functor and let R : C ′′ → C bea right adjoint of F and let A = R ( ). Then there exists a half-braiding σ on A such that ( A, σ ) is a commutative algebra in the Drinfeld center Z ( C ) of C , whichsatisfies Hom C ( , A ) ∼ = k . The algebra ( A, σ ) is called the induced central algebraof F [8, Section 6].The category C A or right A -modules in C becomes a tensor category with tensorproduct ⊗ A and unit object A , equipped with a tensor functor F A =? ⊗ A : C → C A .Furthermore, there is an equivalence of tensor categories κ : C ′′ → C A such thatthe following diagram of tensor functors is commutative up to a monoidal naturalisomorphism C F / / F A ❆❆❆❆❆❆❆❆ C ′′ κ (cid:15) (cid:15) C A . XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 13
For instance, let H be a finite dimensional Hopf algebra. Then the inducedcentral algebra ( A, σ ) of the forgetful functor comod − H → Vec k can be describedas follows: As an algebra in comod − H , A = H with the right regular coaction∆ : A → A ⊗ H . For any right H -comodule V , the half-braiding σ V : A ⊗ V → V ⊗ A is determined by the right adjoint action of H in the form σ V ( h ⊗ v ) = v (0) ⊗ S ( v (1) ) h v (2) , h ∈ H, v ∈ V. See [8, Example 6.3].Let C be a tensor category. We shall say that an algebra ( A, σ ) in the center Z ( C ) is a central algebra of C if ( A, σ ) is the induced central algebra of somedominant perfect tensor functor. Thus a central algebra of C is a commutativealgebra ( A, σ ) ∈ Z ( C ) such that Hom C ( , A ) is one-dimensional. The dominanttensor functor corresponding to ( A, σ ) is the functor − ⊗ A : C → C A , whose rightadjoint is the forgetful functor C A → C . Definition 4.6.
An exact sequence sequence of tensor categories C ′ −→ C F −→ C ′′ is called a perfect exact sequence if F is a perfect tensor functor.Every exact sequence of finite tensor categories is a perfect exact sequence. Inaddition, in every perfect exact sequence C ′ −→ C F −→ C ′′ , the kernel C ′ is a finitetensor category; see [8, Proposition 3.15]. Thus the induced Hopf algebra of aperfect exact sequence is always finite dimensional.Let ( E ) : C ′ −→ C −→ C ′′ be a perfect exact sequence and let ( A, σ ) ∈ Z ( C ) bethe induced central algebra of F . Then ( A, σ ) is self-trivializing , that is, A ⊗ A isa trivial object of C A . Let h A i be the smallest abelian subcategory of C containing A and stable under direct sums, subobjects and quotients. Then F A : C → C A is anormal dominant tensor functor with Ker F A = h A i . Moreover, ( E ) is equivalent tothe exact sequence(4.4) h A i −→ C F A −→ C A . See [8, Subsection 6.2].4.4.
Examples from finite groups: equivariantization and crossed exten-sions by matched pairs.
Let G be a finite group and let C be a tensor category.A monoidal functor ρ : G op → Aut k ( C ) is called a right action of G on C by k -linearautoequivalences.The equivariantization of C under the action ρ is the k -linear abelian category C G whose objects are pairs ( X, r ), where X is an object of C and r = ( r g ) g ∈ G is acollection of isomorphisms r g : ρ g ( X ) → X , g ∈ G , such that for all g, h ∈ G ,(4.5) r g ρ g ( r h ) = r hg ( ρ g,h ) X , where ρ g,h : ρ g ρ h → ρ hg is the monoidal structure of ρ , and r e ρ X = id X . Amorphism f : ( X, r ) → ( Y, r ′ ) is a morphism f : X → Y in C such that f r g = r ′ g ρ g ( f ), for all g ∈ G .The forgetful functor F : C G → C gives rise to a perfect exact sequence of tensorcategories(4.6) Rep G −→ C G F −→ C , with induced Hopf algebra H ∼ = k G [8, Subsection 5.3]. An exact sequence of tensorcategories equivalent to (4.6) is called an equivantization exact sequence . A G -grading of C is a decomposition C = L g ∈ G C g into abelian subcategories C g , called the homogeneous components of the grading, such that C g ⊗ C h ⊆ C gh , for all g, h ∈ G . The neutral homogeneous component C e is a tensor subcategoryof C . A G -grading C = L g ∈ G C g is faithful if C g = 0, for all g ∈ G .Every finite tensor category C has a faithful universal grading C = L u ∈ U ( C ) C u ,with neutral homogeneous component C e equal to the adjoint subcategory C ad , thatis, the smallest tensor Serre subcategory of C containing the objects X ⊗ X ∗ , where X runs over the simple objects of C . The group U ( C ) is called the universal gradinggroup of C .The upper central series of C , · · · ⊆ C ( n +1) ⊆ C ( n ) ⊆ · · · ⊆ C (0) = C , is definedas C (0) = C and C ( n +1) = ( C ( n ) ) ad , for all n ≥
0. A tensor category C is called nilpotent if there exists some n ≥ C ( n ) ∼ = Vec. See [16, Section 3.5 and4.14], [21].Let C be a tensor category and let ( G, Γ) be a matched pair of groups. A ( G, Γ)- crossed action on C consists of the following data: • A Γ-grading C = L s ∈ Γ C s . • A right action of G by k -linear autoequivalences ρ : G op → Aut k ( C ) suchthat(4.7) ρ g ( C s ) = C s ⊳ g , ∀ g ∈ G, s ∈ Γ , • A collection of natural isomorphisms γ = ( γ g ) g ∈ G :(4.8) γ gX,Y : ρ g ( X ⊗ Y ) → ρ t ⊲ g ( X ) ⊗ ρ g ( Y ) , X ∈ C , t ∈ Γ , Y ∈ C t , • A collection of isomorphisms γ g : ρ g ( ) → , g ∈ G .These data are subject to the commutativity of the following diagrams:(a) For all g ∈ G , X ∈ C , s, t ∈ Γ, Y ∈ C s , Z ∈ C t , ρ g ( X ⊗ Y ⊗ Z ) γ gX ⊗ Y,Z / / γ gX,Y ⊗ Z (cid:15) (cid:15) ρ t ⊲ g ( X ⊗ Y ) ⊗ ρ g ( Z ) γ t ⊲ gX,Y ⊗ id ρg ( Z ) (cid:15) (cid:15) ρ st ⊲ g ( X ) ⊗ ρ g ( Y ⊗ Z ) id ρst ⊲ g ( X ) ⊗ γ gY,Z / / ρ s ⊲ ( t ⊲ g ) ( X ) ⊗ ρ t ⊲ g ( Y ) ⊗ ρ g ( Z )(b) For all g ∈ G , X ∈ C , ρ g ( X ) ⊗ ρ g ( ) id ρg ( X ) ⊗ γ g ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ρ g ( X ) γ gX, o o = (cid:15) (cid:15) γ g ,X / / ρ g ( ) ⊗ ρ g ( X ) γ g ⊗ id ρg ( X ) w w ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ρ g ( X ) XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 15 (c) For all g, h ∈ G , X ∈ C , s ∈ Γ, Y ∈ C s , ρ g ρ h ( X ⊗ Y ) ρ g ( γ hX,Y ) (cid:15) (cid:15) ρ g,hX ⊗ Y / / ρ hg ( X ⊗ Y ) γ hgX,Y (cid:15) (cid:15) ρ s ⊲ hg ( X ) ⊗ ρ hg ( Y ) ρ g ( ρ s ⊲ h ( X ) ⊗ ρ h ( Y )) γ gρs ⊲ h ( X ) ,ρh ( Y ) / / ρ ( s ⊳ h ) ⊲ g ρ s ⊲ h ( X ) ⊗ ρ g ρ h ( Y ) ρ s ⊳ h ) ⊲ g,s ⊲ hX ⊗ ρ g,hY O O (d) For all g, h ∈ G , ρ g ρ h ( ) ρ g ( γ h ) (cid:15) (cid:15) ( ρ g,h ) / / ρ hg ( ) γ hg (cid:15) (cid:15) ρ g ( ) γ g / / (e) For all X ∈ C , s ∈ Γ, Y ∈ C s , X ⊗ Y ρ X ⊗ ρ Y ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ρ X ⊗ Y / / ρ e ( X ⊗ Y ) γ eX,Y (cid:15) (cid:15) ρ e ( X ) ⊗ ρ e ( Y ) = ❍❍❍❍❍❍❍❍❍❍❍❍❍ ρ / / ρ e ( ) γ e (cid:15) (cid:15) We say that a tensor category C is a ( G, Γ) -crossed tensor category if it isequipped with a ( G, Γ)-crossed action. In this case there is a canonical exact se-quence of tensor categories(4.9) Rep G −→ C ( G, Γ) F −→ C , where C ( G, Γ) is the tensor category defined as follows: As a k -linear category, itis the equivariantization C G of C under the action ρ , while the tensor product isdefined in the form ( X, r ) ⊗ ( Y, r ′ ) = ( X ⊗ Y, ˜ r ), where ˜ r g , g ∈ G , is the composition M s ∈ Γ ρ g ( X ⊗ Y s ) ⊕ s γ gX,Ys −→ M s ∈ Γ ρ s ⊲ g ( X ) ⊗ ρ g ( Y s ) ⊕ s r s ⊲ g ⊗ r ′ gs −→ M s ∈ Γ X ⊗ Y s ⊳ g = X ⊗ Y, for Y = L s ∈ Γ Y s , Y s ∈ C s . The functor F : C ( G, Γ) → C is the forgetful functor F ( V, ( r g ) g ∈ G ) = V . See [43, Theorem 6.1].We call C ( G, Γ) a ( G, Γ) -crossed extension of C e . Thus, a ( G, Γ)-crossed extensionis a unified formulation of equivariantizations and group graded extensions. In fact,suppose that ρ : G op → Aut ⊗ ( C ) is an action by tensor auto-equivalences of a tensorcategory C . Then the equivariantization C G is a ( G, { e } )-crossed extension of C ,where { e } is the trivial group endowed with the trivial actions ⊳ : { e } × G → G and ⊲ : { e } × G → { e } . On the other hand, if C is a tensor category graded by agroup Γ, then C is a ( { e } , Γ)-crossed extension of C e in a similar way.Further examples of ( G, Γ)-crossed extensions are the categories of representa-tions of abelian extensions of Hopf algebras. Indeed for abelian every exact sequence k −→ k Γ −→ H −→ kG −→ k , the category H -mod is a ( G, Γ)-crossed extensionof Vec. See [43, Subsection 8.2].There exist ( G, Γ)-crossed extensions that cannot be built up by means of equiv-ariantizations or group graded extensions. For instance, let n ≥ H = k A n − kC n , n ≥
5, be the bicrossed product associated to thematched pair ( C n , A n − ) arising from the exact factorization A n = A n − C n of thealternating group A n , where C n = h (12 . . . n ) i [31, Section 8].As shown in [47, Example 4.3], if G is a nontrivial finite group, then H -mod isnot equivalent to a G -equivariantization or to a G -graded extension of any fusioncategory C .4.5. Abelian exact sequences of tensor categories and matched pairs ofgroups.Definition 4.7. ([47, Definition 5.1].) An exact sequence of tensor categories isan abelian exact sequence if its induced Hopf algebra H is finite dimensional andcommutative.Equivalently, an abelian exact sequence is an exact sequence of the formRep G −→ C −→ C ′′ , such that the induced tensor functor ω : Rep G → Vec is monoidally isomorphicto the forgetful functor or, in other words, such that the corresponding rank-onemodule category M is equivalent to the trivial rank-one module category of Rep G .Examples of abelian exact sequences of tensor categories arise from equivarianti-zation under the action of a finite group on a tensor category and also from Hopfalgebra extensions of the form k −→ k G −→ H −→ H ′′ −→ k, where G is a finite group.Let ( G, Γ) be a matched pair of finite groups and let C be a ( G, Γ)-crossed tensorcategory. The induced Hopf algebra of the associated exact sequence of tensorcategories(4.10) Rep G −→ C ( G, Γ) F −→ C is H ∼ = k G , so that (4.9) is an abelian exact sequence. The main result of [47] saysthat crossed extensions by matched pairs do in fact exhaust the class of abelianexact sequences of finite tensor categories: Theorem 4.8. ([47, Theorem 1.1].)
Let G be a finite group and let ( E ) : Rep G −→ C −→ D be an abelian exact sequence of finite tensor categories. Then there exists a finitegroup Γ endowed with mutual actions by permutations ⊲ : Γ × G → G , ⊳ : Γ × G → Γ and a ( G, Γ) -crossed action on D such that ( E ) is equivalent to the exact sequence Rep G −→ D ( G, Γ) −→ D . Exact sequences of finite tensor categories with respect to amodule category
Let C and D be finite k -linear abelian categories. Their Deligne tensor product is a finite tensor category denoted C ⊠ D endowed with a functor ⊠ : C × D → C ⊠ D exact in both variables such that for any k -bilinear right exact functor F : C × D →A , where A is a k -linear abelian category, there exists a unique right exact functor˜ F : C ⊠ D → A such that ˜ F ◦ ⊠ = F . Such a category exists and it is unique up XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 17 to equivalence. In fact, if
C ∼ = A -mod and D ∼ = B -mod, for some finite dimensional k -algebras A and B , then C ⊠ D ∼ = ( A ⊗ B )-mod. See [11]. The tensor product oftwo finite (multi-)tensor categories C and D is again a finite tensor category and if C and D are (multi-)fusion categories, then so is C ⊠ D .Let Y, Z be objects of C and D , respectively, and let us denote Y ⊠ Z = ⊠ ( Y, Z ).Then Hom C ⊠ D ( Y ⊠ Z , Y ⊠ Z ) ∼ = Hom C ( Y , Y ) ⊗ Hom D ( Z , Z ), for all Y , Y ∈ C , Z , Z ∈ D . The simple objects of C ⊠ D are exactly those of the form Y ⊠ Z , where Y is a simple object of C and Z is a simple object of D .Let C ′ ⊆ C and C ′′ be finite tensor categories and let M be an exact indecompos-able left C ′ -module category. In particular, M is finite. Let End( M ) denote the cat-egory of k -linear right exact endofunctors of M , which is a monoidal category withtensor product given by composition of functors and unit object 1 M : M → M .Let also i : C ′ → C denote the inclusion functor. Definition 5.1.
An exact sequence of tensor categories with respect to M is asequence of exact monoidal functors(5.1) C ′ i −→ C F −→ C ′′ ⊠ End( M ) , such that F is dominant, C ′ = Ker F coincides with the subcategory of C mappedto End( M ) under F and, F is normal in the sense that for every object X of C ,there exists a subobject X of X such that F ( X ) is the largest subobject of F ( X )contained in End( M ).A tensor category C fitting into an exact sequence 5.1 with respect to M is calledan extension of C ′′ by C ′ with respect to M .The notion of exact sequence with respect to a module category was introduced in[14] and it generalizes the notion of exact sequence of [8]. Indeed, suppose that (5.1)is an exact sequence of finite tensor categories with respect to M . Then FPdim C =FPdim C ′ FPdim C ′′ . Moreover this condition characterizes the exactness of (5.1)under the assumptions that F is dominant and C ′ ⊆ Ker F [14, Theorem 3.6].Consider an exact sequence of finite tensor categories(5.2) C ′ −→ C −→ C ′′ , as introduced in Section 4.1. Then (5.2) induces a fiber functor ω : C ′ → Vec, thusmaking M = Vec into a rank-one C ′ -module category. In this way (5.2) becomesan exact module category with respect to the rank-one module category M .The Deligne tensor product C ′′ ⊠ C ′ of two finite tensor categories gives rise to anexact sequence (5.1) with respect to any exact indecomposable C ′ -module category M , where F : C ′′ ⊠ C ′ → C ′′ ⊠ End( M ) is the natural dominant monoidal functor.[14].The notion of exact sequence with respect to a module category is self-dual inthe following sense: Let N be an indecomposable exact C ′′ -module category. Then(5.1) induces an exact sequence with respect to N :(5.3) ( C ′′ ) ∗N F ∗ −→ C ∗N ⊠ M i ∗ −→ ( C ′ ) ∗M ⊠ End( N ) . Suppose that (5.1) is an exact sequence with respect to M . Observe that if C isa fusion category, then C ′ and C ′′ are fusion categories and M is a finite semisimple C ′ -module category.Assume conversely that C ′ and C ′′ are fusion categories. In particular M is afinite semisimple module category over C ′ . By [14, Theorem 3.8] C is also a fusioncategory. In this case the monoidal category End( M ) ∼ = ∨ M ⊠ M is a multifusioncategory. Here, ∨ M is the right C ′ -module category such that ∨ M = M and M ¯ ⊗ X = ∗ X ⊗ M , X ∈ C ′ , M ∈ M ; that is, if M ∼ = C ′ A is the category of right A -modules in C ′ for some indecomposable algebra A ∈ C ′ , then ∨ M ∼ = A C ′ . Lemma 5.2.
Let C ′ i −→ C F −→ C ′′ ⊠ End( M ) be an exact sequence with respect to M . If C is a pointed fusion category, then so is C ′′ .Proof. By assumption every simple object X of C is invertible, that is, X ⊗ X ∗ ∼ = .Let 1 M = L i ∈ I i be a decomposition of the unit object 1 M of End( M ) into adirect sum of simple subobjects 1 i , i ∈ I . Then 1 i ⊗ j ∼ = δ i,j i , and 1 ∗ i ∼ = ∗ i ∼ = 1 i ,for all i, j ∈ I ; see [16, Section 4.3].Let Y be a simple object of C ′′ . For each fixed i ∈ I , Y ⊠ i is a simple objectof C ′′ ⊠ End( M ). Since the functor F : C → C ′′ ⊠ End( M ) is dominant, then Y ⊠ i is a direct summand of F ( X ) for some simple object X of C . Therefore( Y ⊠ i ) ⊠ ( Y ⊠ i ) ∗ ∼ = ( Y ⊗ Y ∗ ) ⊠ i is a direct summand of F ( X ) ⊗ F ( X ) ∗ . Onthe other hand, F ( X ⊗ X ∗ ) ∼ = F ( ) ∼ = C ⊠ End( M ) ∼ = ⊠ M ∼ = M i ∈ I ⊠ i . Therefore the only simple constituent of Y ⊗ Y ∗ is the trivial object of C ′′ . Thisimplies that Y ⊗ Y ∗ ∼ = , that is, Y is invertible. Since the simple object Y wasarbitrary, this shows that C ′′ is pointed, as claimed. (cid:3) Exact sequences and exact factorizations.
Let C be a fusion category.Recall from [20] that an C is endowed with an exact factorization into a product oftwo fusion subcategories A and B if the following conditions hold: • C coincides with the full abelian subcategory spanned by direct summandsof X ⊗ Y , X ∈ A , Y ∈ B , • A ∩ B = h i .If this holds, we write C = A • B .By [20, Theorem 3.8], C = A • B if and only if every simple object Z of C admitsa decomposition Z ∼ = X ⊗ Y , for unique (up to isomorphism) simple objects X ∈ A , Y ∈ B .Thus the notion of an exact factorization can be formulated as a category equiv-alence as follows: C has an exact factorization C = A • B if and only if the tensorproduct of C induces an equivalence of k -linear categories ⊗ : A ⊠ B → C . For example, let E be a finite group. Exact factorizations of E correspond exactlyto exact factorizations of the category Vec E . In fact, if Γ and G are subgroups of E , then E = Γ G is an exact factorization of E if and only if Vec E = Vec Γ • Vec G isan exact factorization of Vec E .Exact factorizations and extensions are related as follows: Every exact sequence C ′ i −→ C F −→ C ′′ ⊠ End( M ) with respect to M induces an exact factorization of thedual fusion category C ∗C ′′ ⊠ M :(5.4) C ∗C ′′ ⊠ M = C ′′ • ( C ′ ) ∗M . Conversely, every exact factorization C = A • B of a fusion category C inducesan exact sequence, with respect to any indecomposable A -module category N (5.5) A ∗N −→ C ∗B ⊠ N −→ B ⊠ End( N ) . In particular, C fits into an exact sequence with respect to N = A :(5.6) A −→ C −→ B ⊠ End( A ) . See [20, Theorem 4.1].
XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 19
Exact factorizations arising from exact sequences of finite tensorcategories.
Consider an exact sequence of Hopf algebras(5.7) k −→ H ′ −→ H −→ H ′′ −→ k, such that H ′ is finite dimensional. Then H is free as a left (or right) module over H ′ and in particular the sequence is cleft [53, Theorem 2.1 (2)]. By [8, Proposition3.9] the exact sequence (5.7) gives rise to an exact sequence of tensor categories(5.8) comod- H ′ → comod- H → comod- H ′′ . Observe that H ′ is naturally an algebra in comod- H . By a result of [51], thetensor category H ′ (comod- H ) H ′ is equivalent to the category of comodules overa certain coquasibialgebra (( H ′ ) ∗ ⊲⊳ H ′′ , ϕ ), where ϕ is a so-called Kac 3-cocycle associated to the exact sequence (5.7) (see [51, Section 6]). The coquasibialgebra(( H ′ ) ∗ ⊲⊳ H ′′ , ϕ ), termed a generalized product coquasibialgebra in [51], has an exactfactorization into its sub-coquasibialgebras ( H ′ ) ∗ and H ′′ .A generalization of this feature to the context of exact sequences of finite tensorcategories appeared in [34]. Let ( E ) : C ′ −→ C −→ C ′′ be an exact sequenceof finite tensor categories and let ( A, σ ) ∈ Z ( C ) be its induced central algebra.So that C ′′ ∼ = C A and the exact sequence ( E ) is equivalent to the exact sequence h A i −→ C F A −→ C A . See Subsection 4.3.Let A C A be the tensor category of A -bimodules in C . The category C ′′ ∼ = C A isan indecomposable exact C -module category and there are equivalences of tensorcategories C ∗C ′′ ∼ = A C A , H -mod ∼ = A (comod − H ) A ∼ = A C ′ A , where H is the induced Hopf algebra of ( E ).It was shown in [34, Proposition 7.3] that there is an equivalence of k -linearcategories(5.9) C ∗C ′′ ∼ = ( H -mod) ⊠ C ′′ , thus, in the sense of the formulation explained at the beginning of this section, anexact factorization of C ∗C ′′ .More precisely, under the identifications H -mod ∼ = A (comod − H ) A ∼ = A C ′ A ⊆ A C A and C ′′ ∼ = C A ⊆ A C A , the tensor product functor ⊗ A : A C ′ A ⊠ C A → A C A induces anequivalence of k -linear categories(5.10) ( H -mod) ⊠ C A −→ A C A . Notice that in the situation of the exact sequence (5.8), there is an equivalenceof tensor categories (comod- H ) ∗ comod - H ′′ ∼ = H ′ (comod- H ) H ′ . See [34, Example 7.4]. 6.
Examples and open questions
We begin this section by discussing some examples that answer a number ofnatural questions regarding the behaviour of exact sequences in relation with knownfacts about exact sequences of groups and Hopf algebras.
Let C be a finite tensor category. Let also C ′ be a tensor subcategory of C and M an indecomposable exact C ′ -module category. We shall say that C ′ is normal in C with respect to M if there exist a tensor category C ′′ and an exact sequence oftensor categories with respect to M : C ′ −→ C −→ C ′′ ⊠ End( M ) . A tensor subcategory C ′ will be called normal in C if it is normal with respectto a rank-one module category, that is, if there exist a tensor category C ′′ and anexact sequence of tensor categories in the sense of Definition 4.1: C ′ −→ C −→ C ′′ . A tensor category C will be called simple if it has no normal tensor subcategorieswith respect to a module category. If C has no normal tensor subcategories it willbe called simple with respect to rank-one module categories .6.1. Fusion subcategories of index 2 need not be normal.
It is a well-knownfact that if G is a finite group, every subgroup of G whose index is the smallest primenumber dividing the order of G is normal in G . More generally, if H is a semisimpleHopf algebra and H ′ is a Hopf subalgebra of H such that dim H = p dim H ′ , where p is the smallest prime number dividing the dimension of H , then H ′ is normalin H [27]. The next theorem gives a generalization of this result in the context offusion categories.Let F : C → D be a dominant tensor functor between finite tensor categories.The
Frobenius-Perron index of F is defined as the ratio FPdim C / FPdim D . TheFrobenius-Perron index of a dominant tensor functor is an algebraic integer, by[17, Corollary 8.11]. In addition, if G is a left (or right) adjoint of F , then theFrobenius-Perron index of F coincides with FPdim G ( ). See [8, Section 4]. Theorem 6.1. ([8, Proposition 4.13], [9, Theorem 6.2]).
Let F : C → D be adominant tensor functor between fusion categories C and D . Then the followinghold:(i) If the Frobenius-Perron index of F is 2, then F is normal.(ii) If C has integer Frobenius-Perron dimension and the Frobenius-Perron indexof F is the smallest prime number dividing FPdim C , then F is normal.Furthermore, the exact sequences arising from (i) and (ii) are equivariantizationexact sequences. Let C be a finite tensor category and let C ′ ⊆ C be a tensor subcategory. The ratioFPdim C / FPdim C ′ will be called the index of C ′ (in C ). This is also an algebraicinteger [17].The dual statement of Theorem 6.1 is not true, that is, there exist fusion subcat-egories of index 2 which are not normal. These examples are Tambara-Yamagamifusion categories T Y ( Z p , χ, τ ) of Frobenius-Perron dimension 2 p , where p is a primenumber. In this case the pointed subcategory of T Y ( Z p , χ, τ ), which is the uniquefusion subcategory of Frobenius-Perron dimension p , is not normal [9, Proposition6.3].In fact, Tambara-Yamagami categories T Y ( Z p , χ, τ ) provide examples of fusioncategories of dimension 2 p which are simple. The next proposition generalizes [9,Proposition 6.5] to the context of exact sequences with respect to a module category. Proposition 6.2.
The fusion category
T Y ( Z p , χ, τ ) is simple. XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 21
Proof.
Let C = T Y ( Z p , χ, τ ). Suppose on the contrary that there is an exactsequence C ′ −→ C −→ C ′′ ⊠ End( M ), for some proper fusion subcategory C ′ andsome indecomposable C ′ -module category M . Then FPdim C = FPdim C ′ FPdim C ′′ and therefore FPdim C , FPdim C ′ ∈ { , p } .Consider the associated exact factorization C ∗C ′′ ⊠ M = C ′′ • ( C ′ ) ∗M . Since every fu-sion category of prime Frobenius-Perron dimension is pointed [17], then C ′′ , ( C ′ ) ∗M are pointed and therefore so is C ∗C ′′ ⊠ M . Thus C is group-theoretical. This is impos-sible because C is not integral (in fact the unique non-invertible simple object of C has Frobenius-Perron dimension √ p ). This contradiction shows that such an exactsequence cannot exist and therefore C is simple, as claimed. (cid:3) Further examples of simple fusion categories of dimension p a q b . Propo-sition 6.2 implies that Burnside’s p a q b -theorem does not extend to fusion categoriesin terms of exact sequences with respect to module categories. Further examplesof this situation are provided by the non-group-theoretical fusion categories con-structed in [24].Let p < q be prime numbers such that p is odd and divides q + 1. Let also ζ = ζ ∈ F q such that ζ p = ζ p = 1 but ζ ζ = 1, and let ξ ∈ H ( Z p , k × ) ∼ = Z p .Consider the non-group-theoretical fusion category C ( p, q, { ζ , ζ } , ξ ) constructedin [24]. The fusion categories C ( p, q, { ζ , ζ } , ξ ) are Z p -extensions of Vec Z q × Z q andthey fall into ( p − p ) / C ( p, q, { ζ , ζ } , ξ ) = pq . Proposition 6.3.
The fusion category C ( p, q, { ζ , ζ } , ξ ) is simple.Proof. As in the proof of Proposition 6.2, let us assume on the contrary that there isan exact sequence C ′ −→ C −→ C ′′ ⊠ End( M ), for some proper fusion subcategory C ′ and some indecomposable C ′ -module category M . So that FPdim C ′′ can beeither p , q , q or pq and similarly for FPdim C ′ . Therefore both C ′′ and ( C ′ ) ∗M arepointed: this follows from [17, Corollaries 8.30 and 8.31] if the Frobenius-Perrondimensions are p , q or q , and by the classification of fusion categories of dimension pq in [15, Theorem 6.3], since p is odd (and thus it cannot divide q − C ∗C ′′ ⊠ M = C ′′ • ( C ′ ) ∗M is pointed and then C is group-theoretical.This contradicts the choice of C and shows that such an exact sequence cannot exist.Thus C is simple, as claimed. (cid:3) Group-theoretical fusion categories and exact sequences.
As men-tioned before, the category of representations of a finite simple group is a simplefusion category.Notice, however, that there exist simple Hopf algebras H such that the tensorcategory H -mod is not simple; see Subsection 3.5.Recall that a fusion category C is called group-theoretical if it is categoricallyMorita equivalent to a pointed fusion category. Let C be a pointed fusion category,so that there exist a finite group G and a 3-cocycle ω : G × G × G → k × such that C is equivalent to the category Vec ωG of G -graded vector spaces with associativitydetermined by ω .Every indecomposable module category over Vec ωG arises from a pair (Γ , α ),where Γ is a subgroup of G and α : Γ × Γ → k × is a 2-cochain on Γ such that dα = ω | Γ × Γ × Γ . Thus, the restriction ω | Γ represents the trivial cohomology class in H (Γ , k × ). Given such a pair (Γ , α ), the twisted group algebra A (Γ , α ) = k α Γ is an indecomposable algebra in Vec ωG . The (left) module category associated to suchpair (Γ , α ) is the category M (Γ , α ) = (Vec ωG ) A (Γ ,α ) of (right) A (Γ , α )-modules in Vec ωG .The group-theoretical category (Vec ωG ) ∗M (Γ ,α ) is denoted C ( G, ω, Γ , α ).Let G be a finite group and let Γ be a subgroup of G . There is a canonical em-bedding of tensor categories Rep Γ −→ C ( G, ω, Γ , α ). The next proposition impliesthat Rep Γ is not necessarily a normal tensor subcategory of C ( G, ω, Γ , α ).Let G be the alternating group A of order 60 and let Γ be a subgroup of G such that G ∼ = A . Then Γ is a maximal subgroup of A . There are 12 suchsubgroups and they constitute 2 conjugacy classes, represented by the subgroups h (12345) , (123) i and h (1 , , , , , (1 , , i .Let C = C ( A , , Γ , C is a group-theoretical fusion category of dimen-sion 360 which is categorically Morita equivalent to Vec A . In addition, C containsa fusion subcategory C ′ ∼ = Rep A .Observe that C is not a graded extension of any fusion category: this follows fromthe characterization of graded extensions in [18, Proposition 2.9], since Rep A isthe unique Tannakian subcategory of Z ( C ) (c.f. [45, Example 3.3]) and C cannotbe an A -graded extension of a fusion subcategory, since it is not pointed.We have b Γ = 1 and Γ = N A (Γ) (by maximality of Γ). Hence, by [22, Theorem5.2], the group of invertible objects of C is trivial. Proposition 6.4.
The category C ( A , , Γ , is simple with respect to rank-onemodule categories.Proof. Suppose that F : C → D is a normal dominant tensor functor such that F is not an equivalence. Then the restriction of F to C ′ ∼ = Rep A is also a normaltensor functor with kernel Ker F ∩C ′ . Therefore either C ′ ∩ Ker F ∼ = Vec or C ′ ⊆ Ker F .Suppose first that C ′ ∩ Ker F ∼ = Vec. Then the restriction of F to C ′ is a fullembedding, whence the dimension of D is divisible by 60. Then the dimension of Ker F can equal 6, 2 or 3. But this contradicts the fact that C has no nontrivialinvertible objects. Hence C ′ ⊆ Ker F .The Grothendieck ring Gr( C ) of C is faithfully graded by the double coset ringΓ \ A / Γ. In addition the fusion subcategory
Ker F determines a based subring ofGr( C ). Since C ′ ⊆ Ker F , this based subring corresponds to a subgroup of A containing Γ, see [22, Theorem 4.1 and Lemma 3.6]. The maximality of Γ impliesthat C ′ = Ker F . Then C fits into an exact sequence C ′ −→ C −→ D , where D is afusion category of dimension 6. Moreover, by [8, Proposition 4.9], D is integral.Assume that the exact sequence C ′ −→ C −→ D is abelian . In other words thesequence is exact with respect to the trivial C ′ -module category Vec in the senseof [14]. Let ( A, σ ) ∈ Z ( C ) be the induced central algebra of F . Then A C A has afactorization into fusion subcategories equivalent, respectively, to Vec Γ and D . If D is pointed of dimension 6, then D ∼ = Vec ωS , where S is a group of order 6. Hence, A C A ∼ = Vec ω ′ L , where L is a group of order 360 that has an exact factorization L = Γ .H [34, Proposition 7.3]. In particular the pointed fusion category Vec ω ′ L iscategorically Morita equivalent to Vec A . By [37, Theorem 5.8], this implies that L ∼ = A . This leads to a contradiction, because the group A admits no exactfactorization into proper subgroups [33]. XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 23
If, on the other hand, D is not pointed, then D must contain two distinct invert-ible objects and a simple object of dimension 2. Then the group of invertible objectsof A C A is of order 120. Since A C A is Morita equivalent to Vec A , then there exists asubgroup T of A and a 2-cocycle ψ on T such that A C A ∼ = C ( A , , T, ψ ). By [22,Theorem 5.2], the group of invertible objects of A C A has order | K || b T | , where K isa certain subgroup of N A ( T ) /T . A direct inspection on the possible subgroups T of A (see Table 1) shows that | N A ( T ) /T || b T | ≤
36, which is again a contradiction.We have thus shown that the group-theoretical fusion category C = C ( A , , G, abelian exact sequence of tensor categories.Therefore, if C is not simple, then C fits into a non-abelian exact sequence(6.1) C ′ −→ C −→ D , where C ′ ∼ = Rep A and FPdim D = 6. By the previous part, the induced Hopfalgebra H of (6.1) is not commutative. Hence H ∼ = ( k A ) J , where the twist J isnot trivial. Then H -mod is of type (1 ,
12; 4 , C ∗D ∼ = H -mod • D . As before, thefusion category D is either pointed or of type (1 ,
2; 2 , C ∗D are (i) (1 ,
72; 4 , , (ii) (1 ,
24; 2 ,
12; 4 ,
6; 8 , . Write, as before, A C A ∼ = C ( A , , T, ψ ), where T is a subgroup of A and ψ is a 2-cocycle on T . As pointed out before, the group of invertible objects of C ( A , , T, ψ )is at most 36, hence possibility (i) is discarded.If possibility (ii) holds, then T must be a Klein four group (Table 1). Observethat, for every simple object X of C , FPdim X divides | T | ; see [39, Proposition 5.5].This contradicts the fact that C ∗D has simple objects of dimension 8 as in case (ii).This shows that Rep Γ is not normal in C = C ( A , , Γ ,
1) and therefore C is simple,as was to be shown. (cid:3) Dominant images of normal fusion subcategories need not be normal.
Proposition 6.4 provides examples of images of normal fusion subcategories underdominant tensor functors which are not normal.Let G ∼ = A and A ∼ = Γ ⊆ A as in the previous subsection. Let also C = C ( G, , Γ , G gives rise to an embedding of braided tensorcategories Rep G −→ Z ( G ) that fits into an equivariantization exact sequence oftensor categories Rep G −→ Z ( G ) −→ Vec G . On the other hand, since C is Morita equivalent to Rep G , there is an equivalenceof braided tensor categories Z ( G ) ∼ = Z ( C ) [52]. This equivalence induces a dominanttensor functor U Γ : Z ( G ) −→ C , such that U Γ ( V ) = k Γ ⊗ V . In particular U Γ (Rep G ) = Rep Γ. Since, by Proposition6.4, C is simple this implies: Corollary 6.5.
The image of the normal fusion subcategory
Rep G of Z ( G ) underthe dominant tensor functor U Γ : Z ( G ) → C is not normal in C . Composition series and composition factors.
The definition of a com-position series of a finite dimensional Hopf algebra (Definition 3.4) has an obviousextension to the context of exact sequences of finite tensor categories with respectto module categories.Let a composition series of a finite tensor category C be defined as a sequenceof finite tensor categories C , . . . , C n defined, as before, as n = 1 and C = C , if C issimple, while if C ′ −→ C −→ C ′′ ⊠ End( M ) is an exact sequence with respect to the C ′ -module category M such that FPdim C ′ , FPdim C ′′ >
1, and C ′ , . . . , C ′ m , C ′′ , . . . , C ′′ ℓ , are composition series of C ′ and C ′′ , respectively, then n = m + ℓ and C i = ( C ′ i , ≤ i ≤ m, C ′′ i − m , m < i ≤ m + ℓ. As before, the factors and the length of the series are, respectively, the finite tensorcategories C , . . . , C n and the number n .It is clear that every finite tensor category admits such a composition series. Example 6.6.
Let G be a finite simple group. Then the fusion category Rep G is simple: in fact, Rep G has no proper fusion subcategories when G is simple.Observe that this is not true for the category Vec G , since the finite simple group G might have non-trivial exact factorizations into proper subgroups. In particularthe condition on a fusion category being simple is not self-dual.On the other hand, if H is a normal subgroup of G , then the restriction functorgives rise to an exact sequence in the sense of [8]:Rep G/H −→ Rep G −→ Rep H. Inductively, we find that if G , . . . , G n are the composition factors of G , then thefusion categories Rep G , . . . , Rep G n , are composition factors of Rep G .In what follows we show that composition series of fusion categories thus definedfail to satisfy a Jordan-H¨older theorem.Let n ∈ N . We shall denote by S n and A n the symmetric and alternating groupsof degree n , respectively.The proof of the following theorem relies on a result of Miller [33] that assertsthat the alternating group A does not admit any exact factorization into propersubgroups. We summarize in Tables 1 and 2 the information about the subgroupsof A and A used along the proof. Theorem 6.7.
The fusion category
Vec A is simple.Proof. Suppose on the contrary that there exists an exact sequence with respect toan indecomposable C ′ -module category M :(6.2) C ′ −→ Vec A F −→ C ′′ ⊠ End( M ) , such that FPdim C ′ , FPdim C ′′ > C ′ and C ′′ are fusion categories. Since the functor F is dominant then C ′′ is pointed, by Lemma 5.2. Also C ′ is pointed; moreover, C ′ = Vec H , for somesubgroup H of A such that 1 ( H ( A . Hence M = M ( T, ψ ) = C ′ A ( T,c ) , for somesubgroup T of H and some 2-cocycle ψ on T . In addition, 360 = FPdim Vec A =FPdim C ′ FPdim C ′′ = | H | FPdim C ′′ . XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 25
Consider the associated exact factorization of the dual fusion category D =(Vec A ) ∗C ′′ ⊠ End( M ) ∼ = A ( T,ψ ) (Vec A ) A ( T,ψ ) ∼ = C ( A , , T, ψ ):(6.3) D = C ( A , , T, ψ ) = C ( H, , T, ψ ) • C ′′ . Since A admits no exact factorizations, then [37, Theorem 3.4] implies that D is not pointed. Also, since FPdim C ′′ >
1, then FPdim D pt >
1. Moreover C ′′ iscontained in D pt and FPdim C ′′ = [ A : H ]. Also [ A : H ] divides [ A : T ] because T is a subgroup of H .By [22, Theorem 5.2], the group of invertible objects of D is an extension of b T by a certain subgroup K of N A ( T ) /T .Then FPdim D pt divides | N A ( T ) /T || b T | , and therefore [ A : H ] || N A ( T ) /T || b T | .Thus [ A : H ] divides gcd([ A : T ] , | N A ( T ) /T || b T | ). A direct inspection on thepossible subgroups T in Table 1, combined with the fact that A has no subgroupsof index 2, 3 or 4, implies that [ A : H ] = 6 and H ∼ = A .In addition the subgroup T ∼ = Z × Z is a Klein four group, or T ∼ = Z , or T ∼ = A .Suppose first that T ∼ = Z × Z . Then FPdim D pt divides | N A ( T ) /T || b T | = 24.In addition | b T | divides FPdim C ( H, , T, ψ ) pt and therefore 24 divides FPdim D pt .Hence FPdim D pt = 24 = | N A ( T ) /T || b T | . This implies that the subgroup K in [22,Theorem 5.2] coincides with N A ( T ) /T .From the definition of K in [22], we find that for every g ∈ N A ( T ), the class ofthe 2-cocycle ψ g : T × T → k × , defined by ψ g ( h , h ) = ψ ( h , h ) ψ ( g − h − g, g − h − g ) , h , h ∈ T, is trivial. Hence the cocycle ψ g is also trivial for all g ∈ N H ( T ) and thereforeFPdim C ( H, , T, ψ ) pt = | N H ( T ) /T || b T | = 12, since the group N H ( T ) /T is cyclic oforder 3 (see Table 2).Combined with the fact that FPdim C ′′ = 6 and C ′′ is pointed, this implies thatFPdim D pt = 72, and we arrive to a contradiction. Thus we have discarded thispossibility for T .Assume next that T ∼ = Z . In this case the class of ψ is trivial and N A ( T ) /T is of order 6. Thus the subgroup K in [22, Theorem 5.2] coincides with N A ( T ) /T and FPdim D pt = 18. On the other hand, N H ( T ) /T is of order 2 (Table 2) andtherefore FPdim C ( H, , T, ψ ) pt = 6. The exact factorization (6.2) then implies thatFPdim D pt = 36, which is a contradiction. Then this possibility is also discarded.Finally suppose that T ∼ = A . In this case b T is of order 3 and N A ( T ) /T is of order2. Thus FPdim D pt divides 6. But since | b T | also divides FPdim C ( H, , T, ψ ) pt , and C ′′ is pointed then 9 divides FPdim D pt , which is impossible. This discards thispossibility as well. Thus we have shown that such an exact factorization (6.2)cannot exist and therefore Vec A is a simple fusion category, as claimed. (cid:3) Corollary 6.8.
The composition factors of a fusion category may be non-uniqueup to permutation. Moreover, a fusion category may admit composition series withdifferent length.Proof.
The group S has an exact factorization S = A • Z . This induces an exactfactorization Vec S = Vec A • Vec Z . Therefore, from [20, Theorem 4.1], there is anexact sequence with respect to Vec A :(6.4) Vec A −→ Vec S −→ Vec Z ⊠ End(Vec A ) . By Theorem 6.7, Vec A is simple. Then (6.4) gives a composition series for Vec S with factors Vec A , Vec Z . On the other hand, the exact factorizations S = S • Z = S • Z • Z • Z = S • Z • Z • Z • Z = Z • Z • Z • Z • Z • Z = Z • Z • Z • Z • Z • Z , induce exact factorizations of the corresponding pointed fusion categories. An it-erated application of [20, Theorem 4.1] implies that Vec S has a composition serieswith factors Vec Z , Vec Z , Vec Z , Vec Z , Vec Z , Vec Z , Vec Z . Then we see that thefactors, and also the length, of these composition series are not unique. (cid:3) Remark . The statement of Corollary 6.8 remains valid when restricted to theclass of non-degenerate braided fusion categories, namely, also in this case thecompositions factors may be non-unique up to permutation and composition seriesmay have different lengths.As an example, consider the Drinfeld center Z (Vec S ). Thus Z (Vec S ) coincideswith the representation category of the Drinfeld double D ( k S ) and there is an exactsequence of finite dimensional Hopf algebras k −→ k S −→ D ( k S ) −→ k S −→ k. Therefore the non-degenerate braided fusion category Z (Vec S ) fits into an exactsequence in the sense of [8]:Rep S −→ Z (Vec S ) −→ Vec S . In addition we have an exact sequenceRep Z −→ Rep S −→ Rep A , and since Rep A is a simple fusion category, the composition factors of Rep S areRep Z and Rep A .The proof of Corollary 6.8 gives two composition series of Vec S that give rise totwo composition series of Z (Vec S ) with factorsRep Z , Rep A , Vec A , Vec Z , on the one hand, andRep Z , Rep A , Vec Z , Vec Z , Vec Z , Vec Z , Vec Z , Vec Z , Vec Z on the other hand. Then the series have different length and the factors are notunique also in this case.6.6. Questions.
We think it is interesting to determine classes of finite tensorcategories which are closed under extensions. For instance, the class of fusioncategories is closed under extensions and so is the class of weakly integral andintegral finite tensor categories.On the other hand, it is known that the class of group-theoretical fusion cate-gories is not closed under extensions: Indeed, let p be an odd prime number andlet H be one of the non-group-theoretical semisimple Hopf algebras of dimension4 p constructed by D. Nikshych in [48]. Then H fits into an exact sequence of Hopfalgebras k −→ k Z −→ H −→ A p −→ k, where A p is a certain abelian extension of Z by Z p × Z p . XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 27
Automorphism classrepresentative Isomorphismclass of T | T | | b T | [ N G ( T ) : T ] { e } Trivial 1 1 360 h (12)(34) i Z h (12)(34) , (13)(24) i Z × Z h (12)(34) , (12)(56) i Z × Z h (1234)(56) i Z h (1234)(56) , (13)(56) i D h (123) i Z h (123)(456) i Z h (123) , (456) i Z × Z h (123)(456) , (12)(45) i S h (123) , (12)(45) i S h (12)(34) , (123) i A
12 3 2 – h (123)(456) , (14)(25) , (14)(36) i A
12 3 2 h (1234)(56) , (12)(56) i S
24 2 1 h (34)(56) , (12)(56) , (135)(246) , (35)(46) i S
24 2 1 h (123) , (456) , (12)(45) i ( Z × Z ) ⋊ Z
18 2 2 h (123) , (456) , (23)(56) , (14)(2536) i ( Z × Z ) ⋊ Z
36 4 1 h (12345) i Z h (12345) , (25)(34) i D
10 2 1 h (12345) , (123) i A
60 1 1 h (12345) , (14)(56) i A
60 1 1Whole group A
360 1 1
Table 1.
Subgroups T of the alternating group A Automorphism classrepresentative Isomorphismclass of T [ N G ( T ) : T ] { e } Trivial 60 h (12) i Z h (12)(34) i Z × Z h (123) i Z h (123) , (12)(45) i S h (12)(34) , (123) i A h (12345) i Z h (12345) , (25)(34) i D A Table 2.
Subgroups T of the alternating group A Since every abelian extension is group-theoretical [38], then the Hopf algebra A p is group-theoretical. Then H -mod is a non-group-theoretical fusion categorythat fits into an exact sequence of group-theoretical fusion categories A p -mod −→ H -mod −→ Vec Z . A Hopf algebra H is called group-theoretical if the category H -mod (or equivalently, thecategory comod- H ) is group-theoretical. This exact sequence is in fact an equivariantization exact sequence, by constructionof H . See [48, Section 5].A fusion category C is called weakly group-theoretical if it is Morita equivalentto a nilpotent fusion category [18]. The class of weakly group-theoretical fusioncategories is known to be closed under a number of operations, like taking a fusionsubcategory or dominant image, taking Deligne tensor product and Drinfeld center.It is not known if it closed under extensions: Question 6.10. ([20, Question 4.9].)
Is the class of weakly group-theoretical fusioncategories closed under extensions?
It is known that the class of weakly group-theoretical fusion categories is closedunder equivariantizations and group graded extensions [18, Proposition 4.1]. Fur-thermore, it also closed under matched pair crossed extensions [47, Corollary 4.6].Let G be a finite group. In [47] we showed that if C is a fusion category fittinginto an abelian exact sequence Rep G −→ C −→ D or into an exact sequenceVec G −→ C −→ D , then C is weakly group-theoretical if and only if D is weaklygroup-theoretical. As a consequence, every semisolvable semisimple Hopf algebra,as introduced in [36], is weakly group-theoretical.Recall that a fusion category is said to satisfy the Frobenius property if the ratioFPdim C / FPdim X is an algebraic integer, for every simple object X of C . It isknown that every pre-modular fusion category satisfies the Frobenius property [18]. Question 6.11.
Is the class of fusion categories with the Frobenius property closedunder extensions?
It follows from [18, Theorem 1.5] that the class of fusion categories with theFrobenius property is closed under equivariantizations and group graded extensions,and every weakly group-theoretical fusion category satisfies the Frobenius property.The answer to Question 6.11 is not known in general even in the context of Hopfalgebra extensions.Another interesting class of tensor categories is that of
Frobenius tensor cate-gories : these are tensor categories in which every simple object has an injectivehull (equivalently, a projective cover) [4, Subsection 2.3]. For instance, finite tensorcategories and semisimple tensor categories are Frobenius categories.Examples of Frobenius categories are provided by the categories of finite di-mensional comodules over co-Frobenius Hopf algebras: that is, Hopf algebras H endowed with a nonzero integral H → k .It is known that every Hopf algebra H fitting into a strictly exact sequence ofHopf algebras k −→ H ′ −→ H −→ H ′′ −→ k , such that H ′ and H ′′ are co-Frobenius Hopf, is co-Frobenius [3, Theorem 2.10]. This result allows to constructexamples of this kind of Hopf algebras from smaller examples. We do not knownthe answer to the corresponding question for tensor categories: Question 6.12.
Is the class of Frobenius tensor categories closed under extensions?
Regarding the notion of simplicity of a finite tensor category, the following is anatural question, about which very little is known:
Question 6.13.
Is it possible to classify simple (finite) tensor categories?
Finally, motivated by the examples of Subsection 6.5, we ask:
Question 6.14.
Can the definition of a composition series be reformulated in orderthat the Jordan-H¨older theorem holds for finite tensor categories?
XACT SEQUENCES: FROM HOPF ALGEBRAS TO TENSOR CATEGORIES 29
An analogue of the Jordan-H¨older theorem was proved in [44] for weakly group-theoretical fusion categories. The definition of a composition series for this kind ofcategory is given in terms of group equivariantizations and group graded extensionsand the composition factors, which are Morita invariants, are simple finite groups.
References [1] N. Andruskiewitsch,
Notes on extensions of Hopf algebras , Can. J. Math. , 3–42(1996).[2] N. Andruskiewitsch, About finite dimensional Hopf algebras , Contemp. Math. , 1–57(2002).[3] N. Andruskiewitsch, J. Cuadra,
On the structure of (co-Frobenius) Hopf algebras , J.Noncommut. Geom. , 83–104 (2013).[4] N. Andruskiewitsch, J. Cuadra, P. Etingof, On two finiteness conditions for Hopf alge-bras with nonzero integral , Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) , 401–440 (2015).[5] N. Andruskiewitsch, J. Devoto, Extensions of Hopf algebras , Algebra Anal. , 22–61(1995).[6] N. Andruskiewitsch, M. M¨uller, Examples of extensions of Hopf algebras , Rev. Col. Mat. (2015), 193–211.[7] A. Brugui`eres, S. Lack, A. Virelizier, Hopf monads on monoidal categories , Adv. Math. , 745–800 (2011).[8] A. Brugui`eres, S. Natale,
Exact sequences of tensor categories , Int. Math. Res. Not. (24), 5644–5705 (2011).[9] A. Brugui`eres, S. Natale,
Central exact sequences of tensor categories, equivariantizationand applications , J. Math. Soc. Japan , 257–287 (2014).[10] A. Brugui`eres, A. Virelizier, Hopf monads , Adv. Math. , 679–733 (2007).[11] P. Deligne,
Cat´egories tannakiennes , In: The Grothendieck Festschrift, Vol. II, Progr.Math. , 111–195 (1990).[12] V. Drinfeld, S. Gelaki, D. Nikshych and V. Ostrik, On braided fusion categories I , Sel.Math. New Ser. , 1–119 (2010).[13] P. Etingof, S. Gelaki, The representation theory of cotriangular semisimple Hopf alge-bras , Int. Math. Res. Not. , 387–394 (1999).[14] P. Etingof, S. Gelaki,
Exact sequences of tensor categories with respect to a modulecategory , Adv. Math. , 1187–1208 (2017).[15] P. Etingof, S. Gelaki, V. Ostrik, ,
Classification of fusion categories of dimension pq ,Int. Math. Res. Not. (57), 3041–3056 (2004).[16] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor categories , Mathematical Surveysand Monographs , Amer. Math. Soc., Providence, RI, 2015.[17] P. Etingof, D. Nikshych, V. Ostrik,
On fusion categories , Annals of Math (2) , 581–642 (2005).[18] P. Etingof, D. Nikshych, V. Ostrik,
Weakly group-theoretical and solvable fusion cate-gories , Adv. Math , 176–205 (2011).[19] C. Galindo, S. Natale,
Simple Hopf algebras and deformations of finite groups , Math.Res. Lett. (5-6), 943–954 (2007).[20] S. Gelaki, Exact factorizations and extensions of fusion categories , J. Algebra , 505–518 (2017).[21] S. Gelaki, D. Nikshych,
Nilpotent fusion categories , Adv. Math. , 1053–1071 (2008).[22] S. Gelaki, D. Naidu,
Some properties of group-theoretical categories , J. Algebra ,2631–2641 (2009).[23] I. Hofstetter,
Extensions of Hopf algebras and their cohomological description , J. Algebra , 264–298 (1994).[24] D. Jordan, E. Larson,
On the classification of certain fusion categories , J. Noncommut.Geom. , 481–499 (2009).[25] G. I. Kac, Extensions of groups to ring groups , Math. USSR Sbornik (1968), 451–474.[26] G. Karpilovsky, Proyective Representation of Finite Groups , Pure and Applied Mathe-matics , Marcel Dekker, New York-Basel (1985).[27] T. Kobayashi, A. Masuoka, A result extended from groups to Hopf algebras , Commun.Algebra , 1169–1197 (1997).[28] S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf alge-bras by a bicrossproduct construction , J. Algebra , 17–64 (1990).[29] A. Masuoka,
On Hopf algebras with cocommutative coradicals , J. Algebra , 415–466(1991). [30] A. Masuoka,
The p n -th Theorem for Hopf algebras , Proc. Amer. Math. Soc. , 187–195 (1996).[31] A. Masuoka, Extensions of Hopf algebras , Trab. Mat. , FaMAF., Univ. Nac. deC´ordoba, 1999.[32] A. Masuoka,
Hopf algebra extensions and cohomology , Math. Sci. Res. Inst. Publ. ,167-209 (2002).[33] G. A. Miller, Groups which are the products of two permutable proper subgroups , Proc.Acad. USA , 469–472 (1935).[34] M. Mombelli, S. Natale, Module categories over equivariantized tensor categories , Mosc.Math. J. (1), 97–128 (2017).[35] S. Montgomery, Classifying finite dimensional semisimple Hopf algebras, Contemp. Math. , 265–279 (1998).[36] S. Montgomery, S. J. Witherspoon, Irreducible representations of crossed products , J.Pure Appl. Algebra , 315–326 (1998).[37] D. Naidu,
Categorical Morita equivalence for group-theoretical categories , Comm. Alge-bra , 3544–3565 (2007).[38] S. Natale, On group theoretical Hopf algebras and exact factorizations of finite groups ,J. Algebra , 199–211 (2003).[39] S. Natale,
Frobenius-Schur indicators for a class of fusion categories , Pacific J. Math. (2), 353–378 (2005).[40] S. Natale,
Semisolvability of semisimple Hopf algebras of low dimension , Memoirs Amer.Math. Soc. 186, viii+123 pp. (2007).[41] S. Natale,
Semisimple Hopf algebras of dimension 60 , J. Algebra , 3017–3034 (2010).[42] S. Natale,
Jordan-H¨older theorem for finite dimensional Hopf algebras , Proc. Am. Math.Soc. , 5195–5211 (2015).[43] S. Natale,
Crossed actions of matched pairs of groups on tensor categories , TohokuMath. J. (3), 377–405 (2016).[44] S. Natale, A Jordan-H¨older theorem for weakly group-theoretical fusion categories , Math.Z. , 367–379 (2016).[45] S. Natale,
The core of a weakly group-theoretical fusion category , Int. J. Math. , No.2, Article ID 1850012, 23 p. (2018).[46] S. Natale, On the classification of fusion categories , Proc. ICM 2018, World Scientific,Vol , 191–218 (2018).[47] S. Natale, Extensions of tensor categories by finite group fusion categories , Math. Proc.Cambridge Phyl. Soc. to appear. Preprint arXiv:1808.09581 .[48] D. Nikshych, K -rings and twisting of finite-dimensional semisimple Hopf algebras ,Commun. Algebra (1998), 321–342.[49] V. Ostrik, Module categories, weak Hopf algebras and modular invariants , Transform.Groups , 177–206 (2003).[50] B. Parshall, J. P. Wang, Quantum linear groups , Mem. Amer. Math. Soc. , 1991.[51] P. Schauenburg,
Hopf bimodules, coquasibialgebras, and an exact sequence of Kac , Adv.Math. , 194–263 (2002).[52] P. Schauenburg,
The monoidal center construction and bimodules , J. Pure Appl. Algebra , 325–346 (2001).[53] H. -J. Schneider,
Some remarks on exact sequences of quantum groups , Commun. Alg. , 3337–3357 (1993).[54] W. Singer, Extension theory for connected Hopf algebras , J. Algebra , 1–16 (1972).[55] M. Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras , ManuscriptaMath. , 251–270 (1972).[56] M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras , Commun.Algebra , 841–882 (1981).[57] M. Takeuchi, Quotient spaces for Hopf algebras , Commun. Alg. , 2503–2523 (1994). Facultad de Matem´atica, Astronom´ıa, F´ısica y Computaci´on. Universidad Nacionalde C´ordoba. CIEM – CONICET. Ciudad Universitaria. (5000) C´ordoba, Argentina
E-mail address : [email protected] URL: ∼∼