On the trace of the antipode and higher indicators
aa r X i v : . [ m a t h . QA ] O c t ON THE TRACE OF THE ANTIPODE AND HIGHERINDICATORS
YEVGENIA KASHINA, SUSAN MONTGOMERY, AND SIU-HUNG NG
Abstract.
We introduce two kinds of gauge invariants for any finite-dimensionalHopf algebra H . When H is semisimple over C , these invariants are respectively,the trace of the map induced by the antipode on the endomorphism ring of aself-dual simple module, and the higher Frobenius-Schur indicators of the regularrepresentation. We further study the values of these higher indicators in the contextof complex semisimple quasi-Hopf algebras H . We prove that these indicators arenon-negative provided the module category over H is modular, and that for aprime p , the p -th indicator is equal to 1 if, and only if, p is a factor of dim H . Asan application, we show the existence of a non-trivial self-dual simple H -modulewith bounded dimension which is determined by the value of the second indicator. Introduction
Given a Hopf algebra with certain additional structures such as braiding or ribbon,one can define some quantum invariants for knots, links or 3-manifolds (cf. [Tu]).These topological invariants are indeed determined by the monoidal structure of therepresentation category of the underlying Hopf algebra. In general, two Hopf algebraswith inequivalent monoidal categories of their representations may yield two differ-ent quantum invariants. It is then very natural to ask when two finite-dimensional(quasi-) Hopf algebras have equivalent monoidal structures on their representationcategories.The question was first addressed in [S1] for the comodule categories of Hopf alge-bras in terms of Hopf bi-Galois extensions. The same question for the representationcategories of finite-dimensional quasi-Hopf algebras was studied in [EG2] and [NS3].The representation categories of quasi-Hopf algebras H and K are monoidally equiv-alent if, and only if, there exists a gauge transformation F on K such that H isisomorphic, as quasi-bialgebra, to the twist K F of K by F . However, to show theexistence or non-existence of such gauge transformations and isomorphisms remainshighly non-trivial. Therefore, this characterization may not be very practical fordetermining the gauge equivalence of two given quasi-Hopf algebras.Frobenius-Schur (FS) indicators for representations of finite groups were discoveredmore than a century ago. The notion was recently extended to rational conformalfield theory [Ba], to certain C ∗ -fusion categories [FGSV], to semisimple Hopf algebras[LM], to quasi-Hopf algebras [MN], and more generally to pivotal categories [NS2]. Itwas shown in [MN] that the second indicators are invariants of the monoidal structure The second author was supported by NFS grant DMS 07-01291.The third author was supported by NSA grant H98230-08-1-0078. of the representation category of a semisimple quasi-Hopf algebra, and the same resultfor all these indicators was established in [NS3] and [NS2].One of the aims of this paper is to obtain invariants for the monoidal category ofrepresentations of any finite-dimensional Hopf algebra H . In short, these invariantsof H are called gauge invariants . Obviously, the dimension or the quasi-exponent(cf. [EG3]) of a finite-dimensional Hopf algebra over C are gauge invariants. Weapproach this question by re-examining some formulae and properties of FS indicatorsfor semisimple Hopf algebras over C .In [KSZ2], the n -th FS indicator of the regular representation of a semisimple Hopfalgebra H over C with antipode S is given by ν n ( H ) = Tr( S ◦ P n − )where P n is the n -th Sweedler power map defined in Definition 2.1. However, thisexpression is well-defined for any finite-dimensional Hopf algebra H , and we prove inTheorem 2.2 that ν n ( H ) is a gauge invariant of H for all natural number n . Moreover,we establish in Proposition 2.7 that the sequence { ν n ( H ) } n ∈ N of these indicators islinearly recursive, and hence the minimal polynomial for this sequence is also a gaugeinvariant of H . The second indicator ν ( H ) is simply the ordinary trace of S . Thiscan be computed quite handily for Taft algebras and they are sufficient to distinguishtheir gauge equivalence classes as shown in Section 3.The trace of S also induces another gauge invariant. For a self-dual absolutelysimple H -module V , the antipode induces an automorphism S V on H/ ann V . Weproved in Section 4 that Tr( S V ) is also a gauge invariant of H . When H is a complexsemisimple Hopf algebra, Tr( S V ) is equal to the product dim V · ν ( V ) of the dimensionand the second FS indicator of V [LM]. In general, if H is pivotal, we prove here inProposition 4.5 that(0.1) Tr( S V ) = qdim ℓ V · ν ( V )where qdim ℓ V and ν ( V ) are respectively the quantum dimension and the 2nd indi-cator of V relative to any pivotal element of H .In the semisimple case, FS indicators have also been studied extensively in manydifferent contexts (cf. [KMM], [KSZ2], [S3], [NS1]). The explorations carried outin those articles discovered some interesting relations between the dimension of thealgebra, or the category, and the values of these indicators.We learned in group theory that if a finite group G admits a self-dual complexirreducible representation, then | G | must be even. In terms of FS indicators, thismeans if G admits an irreducible representation with non-zero second indicator, then | G | is even. This result was generalized to semisimple Hopf algebras over an alge-braically closed field of characteristic 0 in [KSZ1]. In this paper, we extend this resultin Theorem 5.8 to any semisimple quasi-Hopf algebra over C at any prime degree FSindicator.The value of ν ( H ) or Tr( S ) is of particular interest. In the group case, it countsthe number of involutions in the group and hence it is always positive. We show by anexample in Section 5 that this is not necessarily the case for Hopf algebras. However, N THE TRACE OF THE ANTIPODE AND HIGHER INDICATORS 3 it is worth noting that if the representation category of a semisimple complex quasi-Hopf algebra H is modular, then the FS indicator ν ( H ) is a non-negative integer.In particular, if H is a semisimple factorizable complex Hopf algebra, then Tr( S ) is anon-negative integer and it has the same parity as dim H . The question of whetherTr( S ) = 0 for some semisimple complex Hopf algebra is not settled in this paper.The actual value Tr( S ) of a complex even dimensional semisimple Hopf algebra H can also be used to determine the existence of a non-trivial self-dual irreduciblerepresentation V of H such thatdim V ≤ dim H − | Tr( S ) − | . Moreover, ν ( V ) = 1 if Tr( S ) >
1, and ν ( V ) = − S ) <
1. This result isproved in Theorem 6.1.The paper is organized as follows: We collect some basic definitions, conventionsand results in Section 1. In Section 2, we introduce the gauge invariant ν n ( H ) forany finite-dimensional Hopf algebra H , and study the sequence { ν n ( H ) } n ∈ N of theseinvariants. In particular, ν ( H ) is the trace of the antipode. We compute the invariant ν ( T ) for all Taft algebras T , and the sequence { ν n ( T ) } for T of dimensions 4 and 9 inSection 3. We continue to introduce another gauge invariant γ ( V ) for all absolutelysimple H -modules V in Section 4, and we show that γ ( V ) is the product of thequantum dimension and the 2nd Frobenius-Schur of V when H is a pivotal Hopfalgebra (0.1). In Section 5, we study the values of ν n ( H ) and the positivity of ν ( H )for a complex semisimple (quasi-) Hopf algebra H in relation to its dimension. Finally,in Section 6, we show an application of ν ( H ) in determining the existence of certainself-dual simple submodules of a semisimple Hopf algebra H over C .1. Preliminaries
In this section, we review some basic facts on gauge equivalence of Hopf algebras,and define some terminology and conventions that we will use in the paper. Through-out this paper, the tensor product of two vector spaces
V, W over the base field k will be denoted by V ⊗ W .Let H be a Hopf algebra over the field k with counit ε , comultiplication ∆ andantipode S . We will use the Sweedler notation ∆( h ) = h ⊗ h for h ∈ H with thesummation suppressed. Following [Ka], a gauge transformation of a Hopf algebra H is an invertible element F ∈ H ⊗ such that ( ε ⊗ id)( F ) = (id ⊗ ε )( F ) = 1. One can twist the Hopf algebra H with a gauge transformation F to a quasi-Hopf algebra H F which is the same algebra H with the same counit ε but its comultiplication ∆ F isgiven by ∆ F ( h ) = F ∆( h ) F − for h ∈ H .
The quasi-Hopf algebra H F is an ordinary Hopf algebra if, and only if, the gaugetransformation F satisfies(1.1) 1 ⊗ ∂F = (1 ⊗ F )(id ⊗ ∆)( F )(∆ ⊗ id)( F − )( F − ⊗ . In general, we call a gauge transformation F of a Hopf algebra H a 2-cocycle if F satisfies (1.1). YEVGENIA KASHINA, SUSAN MONTGOMERY, AND SIU-HUNG NG If F = P i f i ⊗ g i is a 2-cocycle of H with inverse F − = P i d i ⊗ e i , then (1.1)implies(1.2) α F β F = 1 , and β F α F = 1 . where(1.3) α F = X j S ( d j ) e j and β F = X i f i S ( g i ) . Moreover, H F is a Hopf algebra with its antipode S F defined by(1.4) S F ( h ) = β F S ( h ) α F The Hopf algebra H F is called a ( Drinfeld ) twist of the Hopf algebra H by the2-cocycle F . It is worth noting that (1.2) does not hold in general if F is not a2-cocycle.Two Hopf algebras H and H ′ over a field k are said to be gauge equivalent if thereexists a 2-cocycle F of H such that H ′ σ ∼ = H F are isomorphic as bialgebras. Theisomorphism σ induces a k -linear equivalence σ ( − ) : H - mod → H ′ - mod as follows:for V ∈ H - mod , σ V = V as k -linear space with the H ′ -action given by h ′ v := σ ( h ′ ) v for all h ′ ∈ H ′ and v ∈ V ,and σ ( f ) = f for any map f in H - mod . The 2-cocycle F defines the natural isomor-phism ξ := (cid:16) σ V ⊗ σ W F · −→ σ ( V ⊗ W ) (cid:17) for any V, W ∈ H - mod . The triple ( σ ( − ) , ξ, id k ) is an equivalence of tensor categories.Conversely, if H, H ′ are finite-dimensional such that H - mod and H ′ - mod areequivalent tensor categories, then it follows by [S1] (or more generally [EG2], [NS3]) H and H ′ are gauge equivalent. We summarize this Hopf algebra version of [NS3,Theorem 2.2] as follows. Theorem 1.1.
Two finite-dimensional Hopf algebras H and H ′ over a field k aregauge equivalent if, and only if, H - mod fin and H ′ - mod fin are equivalent as ( k -linear)tensor categories. Moreover, if a k -linear functor F : H - mod → H ′ - mod defines anequivalence of tensor categories, then there exist a 2-cocycle F of H and a bialgebraisomorphism σ : H ′ → H F such that k -linear equivalences F and σ ( − ) are naturallyisomorphic. (cid:3) A quantity f ( H ) defined for each Hopf algebra H is called a gauge invariant if f ( H ) = f ( H ′ ) for all Hopf algebras H ′ which are gauge equivalent to H . We willintroduce and discuss some gauge invariants in the remainder of this paper.2. Invariance of the n -th indicator ν n ( H )In this section, we introduce a sequence of scalars ν n ( H ) for each finite-dimensionalHopf algebra H over a field k . We obtain a formula for ν n ( H ) in terms of the integralsof H and H ∗ . Moreover, we prove that this sequence { ν n ( H ) } n ∈ N is linearly recursiveand is a gauge invariant of H . In particular, the minimal polynomial p H of thesequence of higher indicators of H is also a gauge invariant. N THE TRACE OF THE ANTIPODE AND HIGHER INDICATORS 5
Let H be a finite-dimensional Hopf algebra over a field k with antipode S , multi-plication m , comultiplication ∆ and counit ε . We define∆ (1) = id H , and ∆ ( n +1) = (id ⊗ ∆ ( n ) ) ◦ ∆for all integers n ≥
1. By the coassociativity of ∆, it is well-known that∆ ( n +1) = (∆ ( n ) ⊗ id) ◦ ∆for all integers n ≥
1. Similarly, we let m (1) = id H , m ( n ) : H ⊗ n → H be the multipli-cation map of H . Note that ∆ ( n +1) defined here is equal to ∆ n in [Sw, p 11]. Definition 2.1.
The n -th Sweedler power P n ( h ) = h [ n ] of an element h ∈ H isdefined as P n ( h ) = m ( n ) ◦ ∆ ( n ) ( h ) for n ≥ , and we set P ( h ) = h [0] = ε ( h )1 . We define ν n ( H ) := Tr( S ◦ P n − ) for each positive integer n , and call it the n -th indicator of H . Obviously, ν ( H ) = 1, and ν ( H ) = Tr( S ). When H is semisimple and k is analgebraically closed field of characteristic zero, the n -th Frobenius-Schur indicator ν n ( V ) is defined for each V ∈ H - mod . The scalar ν n ( H ) coincides with the n -thFrobenius-Schur indicator of the regular representation of H (cf. [KSZ2]). It followsby the works of [NS3] and [MN] that if H ′ is a Hopf algebra and F : H - mod fin → H ′ - mod fin defines an equivalence of tensor categories, then ν n ( V ) = ν n ( F ( V )) forall positive integer n and V ∈ H - mod . Since F ( H ) ∼ = H ′ as H ′ -modules, we have ν n ( H ) = ν n ( H ′ ) for all positive integers n . Therefore, the sequence { ν n ( H ) } n ∈ N is agauge invariant of semisimple Hopf algebras over k .The notion of higher Frobenius-Schur indicators for a module over a general finite-dimensional Hopf algebra H remains unclear. Nevertheless, the following theoremshows the sequence { ν n ( H ) } n ∈ N is a gauge invariant. Theorem 2.2.
Let H a finite-dimensional Hopf algebra over a field k . Then thesequence { ν n ( H ) } n ∈ N is an invariant of the gauge equivalence class of Hopf algebrasof H . If λ ∈ H ∗ is a right integral and Λ ∈ H is a left integral such that λ (Λ) = 1 ,then ν n ( H ) = λ ( S (Λ [ n ] )) for all positive integer n . To prove the theorem, we need to establish a relationship between ∆ ( n ) F and ∆ ( n ) for a given 2-cocycle F of H . Let us define F = 1 H , and F n +1 = (1 ⊗ F n )(id ⊗ ∆ ( n ) )( F )for all integers n ≥
1. In particular, F = (1 ⊗ F )(id ⊗ ∆ (1) )( F ) = F . Lemma 2.3.
Let F be a 2-cocycle of a finite-dimensional Hopf algebra over k . Thenthe following equations hold for all positive integers n : (2.1) F n +1 = ( F n ⊗ ( n ) ⊗ id)( F ) , YEVGENIA KASHINA, SUSAN MONTGOMERY, AND SIU-HUNG NG (2.2) F n ∆ ( n ) ( h ) = ∆ ( n ) F ( h ) F n for all h ∈ H .
Proof.
Both equations obviously hold for n = 1 ,
2, and we proceed to prove theequalities by induction on n . Assume the induction hypothesis (2.2). Let F = P i f i ⊗ g i . Then, for h ∈ H , X ( h ) ,i f i h ⊗ g i h = ∆ F ( h ) F and(id ⊗ ∆ ( n ) F )( F )(1 ⊗ F n ) = X i f i ⊗ ∆ ( n ) F ( g i ) F n = X i f i ⊗ F n ∆ ( n ) ( g i ) = (1 ⊗ F n )(id ⊗ ∆ ( n ) )( F ) = F n +1 . Thus, F n +1 ∆ ( n +1) ( h ) = X ( h ) ,i (1 ⊗ F n )( f i h ⊗ ∆ ( n ) ( g i h ))= X ( h ) ,i ( f i h ⊗ ∆ ( n ) F ( g i h ))(1 ⊗ F n ) = (cid:16) (id ⊗ ∆ ( n ) F )(∆ F ( h ) F ) (cid:17) (1 ⊗ F n )= ∆ ( n +1) F ( h )(id ⊗ ∆ ( n ) F )( F )(1 ⊗ F n ) = ∆ ( n +1) F ( h ) F n +1 . Using the induction assumption, we find F n +2 = (1 ⊗ F n +1 )(id ⊗ ∆ ( n +1) )( F )= (1 ⊗ F n ⊗ (cid:0) (id ⊗ ∆ ( n ) ⊗ id)(1 ⊗ F ) (cid:1) (id ⊗ ∆ ( n +1) )( F )= (1 ⊗ F n ⊗ (cid:0) (id ⊗ ∆ ( n ) ⊗ id)(1 ⊗ F ) (cid:1) (cid:0) (id ⊗ ∆ ( n ) ⊗ id)(id ⊗ ∆)( F ) (cid:1) = (1 ⊗ F n ⊗ ⊗ ∆ ( n ) ⊗ id)((1 ⊗ F )(id ⊗ ∆)( F )) . Here, the third equality is a consequence of the coassociativity of ∆, and the lastequality follows from the fact that ∆ is an algebra map. By (1.1), we have F n +2 = (1 ⊗ F n ⊗ ⊗ ∆ ( n ) ⊗ id)(( F ⊗ ⊗ id)( F ))= ( F n +1 ⊗ ⊗ ∆ ( n ) ⊗ id)(∆ ⊗ id)( F ) = ( F n +1 ⊗ ( n +1) ⊗ id)( F ) . (cid:3) Remark 2.4.
Lemma 2.3 can also be obtained by considering the proof of a coher-ence result of Epstein [Ep] . For the sake of completeness, a direct algebraic proof isprovided for the lemma.
Lemma 2.5.
Let H be a finite-dimensional Hopf algebra over k and F a 2-cocycleof H . Then for any positive integer n and h ∈ H , we have h [ n +1] = m ( n +1) (cid:16) (id ⊗ ∆ ( n ) F ⊗ id) (cid:0) (1 ⊗ F )(1 ⊗ ∆( h ))( F − ⊗ (cid:1)(cid:17) where m ( n +1) denotes the multiplication of H . N THE TRACE OF THE ANTIPODE AND HIGHER INDICATORS 7
Proof.
By Lemma 2.3 and the coassociativity of ∆, we find(id ⊗ ∆ ( n ) F ⊗ id) (cid:0) (1 ⊗ F )(1 ⊗ ∆( h ))( F − ⊗ (cid:1) = (1 ⊗ F n ⊗ (cid:0) (id ⊗ ∆ ( n ) ⊗ id) (cid:0) (1 ⊗ F )(1 ⊗ ∆( h ))( F − ⊗ (cid:1)(cid:1) (1 ⊗ F − n ⊗ ⊗ F n +1 )(1 ⊗ ∆ ( n +1) ( h ))( F − n +1 ⊗ . Note that if Z ∈ H ⊗ ( n +1) is invertible, then m ( n +2) (cid:0) (1 ⊗ Z )( x ⊗ · · · ⊗ x n +2 )( Z − ⊗ (cid:1) = x · · · x n +2 for all x , . . . , x n +2 ∈ H (cf. [NS3, Lemma 4.4]). Therefore, m ( n +1) (cid:0) (1 ⊗ F n +1 )(1 ⊗ ∆ ( n +1) ( h ))( F − n +1 ⊗ (cid:1) = m ( n +1) ◦ ∆ ( n +1) ( h ) = h [ n +1] and so the result follows. (cid:3) Now, we can prove the main result of this section.
Proof of Theorem 2.2.
Let H be a Hopf algebra over a field k with antipode S . Sup-pose Λ is a left integral of H , and λ a right integral of H ∗ such that λ (Λ) = 1. ByRadford’s trace formulae (cf. [R]),Tr( f ) = λ ( S (Λ ) f (Λ )) for any k -linear operator f on H, where ∆(Λ) = Λ ⊗ Λ is the Sweedler notation with the summation suppressed.Thus, we have ν n ( H ) = λ ( S (Λ ) S (Λ [ n − )) = λ ( S (Λ [ n − Λ )) = λ ( S (Λ [ n ] )) . Now let F = P i f i ⊗ g i be a 2-cocycle of H and F − = P j d j ⊗ e j . Then, it followsby (1.2), (1.3) that u = β F = P i f i S ( g i ) is invertible with u − = α F = P i S ( d i ) e i .Moreover, by (1.4), the antipode S F of H F is given by S F ( h ) = uS ( h ) u − . Let P Fn ( h ) denote the n -th Sweedler power of an element h ∈ H F . Then, for n ≥ ν n +1 ( H F ) = λ ( S (Λ ) S F ◦ P Fn (Λ ))= λ ( S (Λ ) uS ( P Fn (Λ )) u − ) = X i,j λ ( S (Λ ) f i S ( g i ) S ( P Fn (Λ )) S ( d j ) e j )= X i,j λ ( S ( S − ( f i )Λ ) S ( d j P Fn (Λ ) g i ) e j ) . Recall from [R] that a Λ ⊗ Λ = Λ ⊗ S − ( a )Λ , Λ a ⊗ Λ = Λ ⊗ Λ S ( a ↼ α )and λ ( ab ) = λ ( S ( b ↼ α ) a ) for all a, b ∈ H YEVGENIA KASHINA, SUSAN MONTGOMERY, AND SIU-HUNG NG where α is the distinguished group-like element in H ∗ defined by Λ a = Λ α ( a ). Usingthese properties, we have ν n +1 ( H F ) = X i,j λ ( S (Λ ) S ( d j P Fn ( f i Λ ) g i ) e j )= X i,j λ ( S ( e j ↼ α ) S (Λ ) S ( d j P Fn ( f i Λ ) g i ) e j )= X i,j λ ( S (Λ S ( e j ↼ α )) S ( d j P Fn ( f i Λ ) g i ))= X i,j λ ( S (Λ ) S ( d j P Fn ( f i Λ e j ) g i ))= X i,j λ ( S ( d j P Fn ( f i Λ e j ) g i Λ )) . Notice that X ij d j P Fn ( f i Λ e j ) g i Λ = m (cid:16) (id ⊗ ∆ ( n ) F ⊗ id) (cid:0) (1 ⊗ F )(1 ⊗ ∆(Λ))( F − ⊗ (cid:1)(cid:17) . Hence, by Lemma 2.5, the last expression is equal to Λ [ n +1] . Therefore, ν n +1 ( H F ) = λ ( S (Λ [ n +1] )) . If H ′ is a Hopf algebra which is gauge equivalent to H , then there exists a 2-cocycle F of H such that H ′ σ ∼ = H F as bialgebras. Let S ′ and P ′ n be the antipode and the n -th Sweedler power map of H ′ respectively. Then σ ◦ S ′ ◦ P ′ n = S F ◦ P Fn ◦ σ . Therefore, ν n ( H ′ ) = ν n ( H F ) = ν n ( H )for all positive integer n . (cid:3) Suppose λ ∈ H ∗ is a right integral and Λ ∈ H is a left integral such that λ (Λ) = 1.Then λ ℓ = λ ◦ S is a left integral of H ∗ and λ ℓ (Λ) = λ ( S (Λ)) = λ (Λ) = 1 (cf. [R]) . Therefore, ν n ( H ) = λ ℓ (Λ [ n ] ).On the other hand, Λ r = S (Λ) is a right integral of H , and we obviously have λ (Λ r ) = 1 , Λ [ n ] r = S (Λ [ n ] ) . Thus, ν n ( H ) = λ (Λ [ n ] r ). We summarize this conclusion in the following corollary. Corollary 2.6.
Let H be a finite-dimensional Hopf algebra over k . Suppose λ ∈ H ∗ and Λ ∈ H are both left integrals (or both right integrals) such that λ (Λ) = 1 . Then ν n ( H ) = λ (Λ [ n ] ) for all positive integer n . (cid:3) N THE TRACE OF THE ANTIPODE AND HIGHER INDICATORS 9
We note, however, that ν n ( H ) is not preserved under twisting by more generalpseudo-cocycles of H . Nikshych [Ni] shows that the group algebras C Q and C D ,where Q is the quaternion group and D is the dihedral group of order 8, are twistsof each other by a pseudo-cocycle. However neither the indicators nor Tr( S ) are thesame for the two groups.Recall that a sequence { β n } n ∈ N in k is said to be linearly recursive if it satisfies anon-zero polynomial f ( x ) = f + f x + · · · + f m − x m − + f m x m ∈ k [ x ], i.e. f a n + f a n +1 + · · · + f m − a n + m − + f m a n + m = 0 for all n ∈ N . The monic polynomial of the least degree satisfied by a linearly recursive sequence iscalled the minimal polynomial of the sequence.For the case of a finite group G , ν n ( C G ) = { g ∈ G | g n = 1 } . Thus, the sequence ν n ( C G ) is periodic, and hence linearly recursive as it satisfies thepolynomial x N − N is the exponent of G . More generally, for any semisimpleHopf algebra H over C , the sequence { ν n ( H ) } n ∈ N is periodic whose period is equalto the exponent of H (cf. [NS1, Proposition 5.3]). However, the example in thefollowing section implies that the sequence of higher indicators { ν n ( H ) } n ∈ N is notperiodic for an arbitrary finite-dimensional Hopf algebra H . Nevertheless, by thefollowing proposition, the sequence is always linearly recursive. Proposition 2.7.
Let H be a finite-dimensional Hopf algebra over any field k . Thenthe sequence { ν n ( H ) } is linearly recursive and the degree of its minimal polynomialis at most (dim H ) . Also, the minimal polynomial p H of the sequence of higherindicators is also a gauge invariant.Proof. Since Hom k ( H, H ) is of finite dimension, the set of operators { P n } n ≥ is k -linearly dependent. There exist a positive integer N ≤ (dim H ) and scalars α , . . . , α N − ∈ k such that(2.3) α P + · · · α N − P N − + P N = 0 . Recall that P n is the n -th power of id H under the convolution product ∗ of Hom k ( H, H ).Therefore, P n ∗ P m = P m + n for any non-negative integers m, n . Multiply Equation(2.3) by P n − . We find(2.4) α P n − + · · · α n − N P n − N + P n − N = 0for all positive integers n . Apply S to this equation and take trace. We have α ν n ( H ) + · · · α n + N − ν n − N ( H ) + ν n + N ( H ) = 0 . Hence, the sequence { ν n ( H ) } is linearly recursive and it satisfies a monic polynomialof degree ≤ N . (cid:3) Indicators of the Taft algebras
The Taft algebras are well-known to be non-semisimple. We will compute the 2ndindicators ν ( T ) of the Taft algebras T as examples in this section. In particular,for the Taft algebra T of dimension 4, we have computed its complete sequence { ν n ( T ) } n ∈ N which is simply the sequence of positive integersLet us begin with the definition of the Taft algebras over an algebraically closedfield k of characteristic zero. For a primitive n -th root of unity ω ∈ k , the Taft algebra T n ( ω ) is the k -algebra generated by g and x subject to the relations g n = 1, x n = 0,and xg = ωgx . The Taft algebra is a Hopf algebra with the coalgebra structure andthe antipode S given by∆( g ) = g ⊗ g, ∆( x ) = x ⊗ g ⊗ x, ε ( g ) = 1 , ε ( x ) = 0 ,S ( g ) = g − , S ( x ) = − g − x . The subset { g i x k | i, k = 0 , . . . , n − } of the Taft algebra T n ( ω ) forms a basis, andhence dim T n ( ω ) = n . By induction, one can write down the images of the antipode S on this basis as Lemma 3.1.
For i, k = 0 , . . . , n − , we have S ( g i x k ) = ( − k ω − ( k ( k − + ik ) g − ( i + k ) x k . By (2.1), ν ( T n ( ω )) = Tr( S ), and so we proceed to compute the trace of S withthe following lemma. Lemma 3.2. If S ( g i x k ) = αg i x k for some α ∈ k , then k ≡ − i (mod n ) . Moreover, Tr( S ) = n − X ℓ =0 ω ℓ = 21 + ω ( n +1) / if n is odd, n − X ℓ =0 ω ℓ = 41 − ω if n is even.Proof. The first assertion follows immediately from Lemma 3.1. Now, let u = (cid:2) n (cid:3) .We first assume n is odd. Then u = n − and so n − u = 1. In particular, u is theinverse of − n . Note that ω / is uniquely determined and is given by ω / = ω − u . Suppose g i x k is an eigenvector of S . Then k ≡ − i (mod n ) and hence i ≡ uk (mod n ). The eigenvalue α associated with this eigenvector is α = ( − k ω − ( k ( k − + ik ) = ( − k ω − ( k ( k − − u )+ uk ) = ( − k ω − uk . Thus, Tr( S ) = n − X k =0 ( − k ω − uk = 1 + ω − un ω − u = 21 + ω − u . N THE TRACE OF THE ANTIPODE AND HIGHER INDICATORS 11
Obviously, − u ≡ ( n + 1) / n ). On the other hand, { ω − uk | k = 0 , . . . , n − } = { ω k | k = 0 , . . . , n − } , and soTr( S ) = X k even ω − uk − X k odd ω − uk = u X ℓ =0 ω ℓ − n − X ℓ = u +1 ω ℓ = 2 u X ℓ =0 ω ℓ . Now, we assume n is even. Then u = n and the congruence k ≡ − i (mod n )implies that k must be even, and so k = 2 ℓ some non-negative integer ℓ < n , and ℓ ≡ − i (mod n . Consequently, i = n − ℓ or n − ℓ . Thus, the eigenvalues associated with the eigenvectors g n − ℓ x ℓ and g n − ℓ x ℓ are ω − ( ℓ (2 ℓ − nℓ − ℓ ) and ω − ( ℓ (2 ℓ − nℓ − ℓ )respectively. However, both eigenvalues are equal to ω ℓ . Note that ω u = −
1. There-fore, Tr( S ) = u − X ℓ =0 ω ℓ = 2 u − X ℓ =0 ω ℓ = 2 (cid:18) − ω u − ω (cid:19) = 41 − ω . (cid:3) It has been shown in [S2, Corollary 2.4] that T n ( ω ) is uniquely determined by itsassociated tensor category T n ( ω )- mod fin . In particular, T n ( ω ) and T n ( ω ′ ) are notgauge equivalent if ω = ω ′ . Here we give an alternative proof by computing the traceof the antipode and using Theorem 2.2. Corollary 3.3.
Let ω , ω ∈ k be primitive n -th roots of unity. Suppose S i is theantipode of the Taft algebra T n ( ω i ) , i = 1 , . Then Tr( S ) = Tr( S ) iff ω = ω . Inparticular, T n ( ω ) and T n ( ω ) are gauge equivalent iff ω = ω .Proof. Suppose Tr( S ) = Tr( S ). If n is even, then, by Lemma 3.2, we have ω = ω .If n is odd, then ω ( n +1) / = ω ( n +1) / . Hence ω = (cid:16) ω ( n +1) / (cid:17) = (cid:16) ω ( n +1) / (cid:17) = ω . The second statement is an immediate consequence of Theorem 2.2. (cid:3)
The general formula for the n -th indicator of T m ( ω ) is less obvious, but T ( −
1) isan exception.
Example 3.4.
The n -th indicator of T ( −
1) is n , and hence its minimal polynomial is( x − . To show this observation, we can apply Corollary 2.6. In T ( − x + gx is a left integral, and λ ∈ T ( − ∗ defined by λ (1) = λ ( g ) = λ ( x ) = 0 and λ ( gx ) = 1is a left integral of T ( − ∗ such that λ (Λ) = 1. By induction, one can show that∆ ( n ) ( x ) = x ⊗ ⊗ ( n − + g ⊗ x ⊗ ⊗ ( n − + · · · + g ⊗ ( n − ⊗ x for n ≥ , where z ⊗ ( ℓ ) denotes the ℓ -folded tensor z ⊗ · · · ⊗ z . Thus, using g = 1,∆ ( n ) ( gx ) = gx ⊗ g ⊗ ( n − + 1 ⊗ gx ⊗ g ⊗ ( n − + · · · + 1 ⊗ ( n − ⊗ gx for n ≥ . Therefore,Λ [ n ] = n − X i =0 g i x + gxg n − − i = n − X i =0 g i x + ( − n − − i g n − i x = n − X i =0 g i x + ( − i g i +1 x . Since λ (Λ [ n ] ) is equal to the coefficient of gx in Λ [ n ] , we find ν n ( T ( − λ (Λ [ n ] ) = n for n ∈ N . (cid:3) A more delicate but direct computation shows that ν n ( T ( ω )) = n (2 + ω ) if n ≡ ,n if n ≡ ,n (1 + ω ) if n ≡ , where ω ∈ k is a primitive third root of unity. The minimum polynomial of thesequence is ( x − ( x − ω − ) . One can continue the computation using GAP to findthe minimal polynomial p m ( x ) of T m ( ω ) for m ≥
3. We summarize our observationfor m ≤
24 as follows: p m ( x ) = ( x m − ( x − ω ) if m = 2 , , , , , , , ( x m − otherwise . It has been shown in [EG3] that the quasi-exponent qexp( H ) of a finite-dimensionalHopf algebra H is a gauge invariant, and qexp( T m ( ω )) = m for all positive integers m . The preceding observation suggests some relation between the quasi-exponentqexp( H ) and the minimal polynomial p H of a finite-dimensional Hopf algebra H . Itwould be interesting to know how they are actually related.4. Gauge invariance of
Tr( S V )Let H be a finite-dimensional Hopf algebra over a field k with antipode S . For V ∈ H - mod fin , we denote the left dual of V ∈ H - mod fin by V ∨ . Suppose V ∈ H - mod fin is self-dual, i.e. V ∼ = V ∨ as H -modules. Then S (ann V ) = ann V and so S induces analgebra anti-automorphism S V on H/ ann V defined by S V ( h +ann V ) = S ( h )+ann V .We define for each absolutely simple V ∈ H - mod fin (4.1) γ ( V ) = (cid:26) Tr( S V ) if V ∼ = V ∨ , γ ( V ) depends only on the isomorphism class of V .In the semisimple case, Tr( S V ) is closely related to the Frobenius-Schur indicator.More precisely, Tr( S V ) = ν ( V ) dim V . It may also prove to be important in thenon-semisimple case. We note that Jedwab has begun the study of Tr( S V ) in [J], andhas computed Tr( S V ) for the irreducible representations of u q ( sl ). Some additionalwork on this topic has been done in [JK]. N THE TRACE OF THE ANTIPODE AND HIGHER INDICATORS 13
Similar to the case of the 2nd Frobenius-Schur indicator of a simple module over asemisimple complex Hopf algebra, we prove in this section that γ ( V ) is an invariantof the tensor category H - mod fin . Theorem 4.1.
Let H , H ′ be finite-dimensional Hopf algebras over a field k . If F : H - mod fin → H ′ - mod fin is a equivalence of tensor categories, then γ ( V ) = γ ( F ( V )) for all absolutely simple V ∈ H - mod fin . When H is pivotal, the 2nd Frobenius-Schur indicator ν ( V ) and pivotal dimensionqdim ℓ ( V ) are defined for each finite-dimensional H -module V . In this case, we provein Proposition 4.5 that γ ( V ) = ν ( V ) qdim ℓ ( V ) = ν ( V ) qdim r ( V )for each absolutely simple H -module V .To prove Theorem 4.1, we first need the following lemma. We thank H.-J. Schneiderfor helpful conversations about the lemma. Lemma 4.2.
Let U = [ U ij ] be an invertible element of M n ( k ) , and A an algebraanti-automorphism of M n ( k ) defined by A ( X ) = U X t U − for some U ∈ GL n ( k ) ,where X t denotes the transpose of X . Then Tr( A ) = tr( U t U − ) where tr is the ordinary trace of matrices.Proof. Let E ij be the matrix [ δ ij ] ∈ M n ( k ), and U − = [ U ij ]. Since h X, Y i = tr( XY t )defines a non-degenerate symmetric bilinear form on M n ( k ) and h E ij , E kl i = δ ij,kl ,we haveTr( A ) = X i,j h A ( E ij ) , E ij i = X i,j tr( U E ji U − E ji ) = X i,j U ij U ij = tr( U t U − ) . (cid:3) The second step is to show that γ ( V ) is invariant under 2-cocycle twisting of H .Let F be a 2-cocycle of H , and V ∈ H - mod fin . We denote by V F the same H -module V but considered as an object in H F - mod fin . Proposition 4.3.
Let H be a finite-dimensional Hopf algebra over k and F a 2-cocycle of H . Then γ ( V ) = γ ( V F ) for all absolutely simple V ∈ H - mod fin .Proof. Let F = P i f i ⊗ g i be a 2-cocycle of H and F − = P j d i ⊗ e j . Then the twist H F of H by F has the antipode S F given by S F ( h ) = uS ( h ) u − where u = P i f i S ( g i ) and u − = P i S ( d j ) e j . Suppose V is an absolutely simpleobject of H - mod fin . The left dual V ∨ in H - mod fin is different from V ∨ F in H F - mod fin , but they are isomorphic as H -modules under the duality transformation (cf. [NS1])˜ ξ : V ∨ → V ∨ F defined by˜ ξ ( f )( v ) = f ( u − v ) for f ∈ V ∗ and v ∈ V .Thus, V is self-dual in H - mod fin if, and only if, V F is self-dual in H F - mod fin . There-fore, γ ( V ) = γ ( V F ) = 0 if V is not self-dual in H - mod fin .Now, let us further assume V is self-dual in H - mod fin . Since V F = V as H -modules, H/ ann V = H/ ann V F φ ∼ = M n ( k )as k -algebras, where n = dim k V . We write h for φ ( h + ann V ) ∈ M n ( k ), and let(4.2) S := φ ◦ S V ◦ φ − and S F := φ ◦ S FV F ◦ φ − . Obviously, both S and S F are algebra anti-automorphisms on M n ( k ), and γ ( V ) = Tr( S ) , and γ ( V F ) = Tr( S F ) . Moreover, there is an invertible matrix U such that(4.3) S ( h ) = S ( h ) = U h t U − . From Equation (4.3) we get the following equalities:(4.4) S F ( h ) = S F ( h ) = uU h t U − u − , S ( h ) = S ( h ) = U ( U − ) t hU t U − for all h ∈ H .In view of Lemma 4.2, γ ( V ) = Tr( S ) = tr( U t U − ) . Thus, by Equations (4.3), (4.4) and Lemma 4.2, we find γ ( V F ) = Tr( S F ) = tr( U t u t U − u − ) = tr( U t U − S ( u ) u − )= X i,j tr( U t U − S ( g i ) S ( d j f i ) e j ) = X i,j tr( g i U t U − S ( d j f i ) e j )= X i,j tr( U t U − S ( d j f i ) e j g i ) = tr( U t U − ) = γ ( V ) . (cid:3) Proof of Theorem 4.1.
By Theorem 1.1, if a k -linear equivalence F : H - mod fin → H ′ - mod fin defines a tensor equivalence, then H and H ′ are gauge equivalence, i.e.there exist a 2-cocycle F of H and a bialgebra isomorphism σ : H ′ → H F . Moreover, F is naturally isomorphic to the k -linear equivalence σ ( − ) : H - mod fin → H ′ - mod fin induced by the algebra isomorphism σ . For any absolutely simple V ∈ H - mod fin , V is self-dual in H - mod fin if, and only if, F ( V ) is self-dual in H ′ - mod fin . Thus, if V isnot self-dual, then γ ( V ) = γ ( F ( V )) = 0.Assume V is self-dual in H - mod fin . Then ann σ V = σ − (ann V F ), and so σ inducesan algebra isomorphism σ : H ′ / ann σ V → H/ ann V F . Let S, S ′ and S F denote theantipodes of H , H ′ and H F respectively. Then σ − ◦ S F ◦ σ = S ′ , N THE TRACE OF THE ANTIPODE AND HIGHER INDICATORS 15 and so σ − ◦ S FV F ◦ σ = S ′ σ V . Therefore, by Proposition 4.3, γ ( F ( V )) = γ ( σ V ) = Tr( S ′ σ V ) = Tr( S FV F ) = γ ( V F ) = γ ( V ) . (cid:3) It is clear that the invariant γ ( V ) of the tensor category H - mod fin is closely relatedto the 2nd Frobenius-Schur indicator when H is split semisimple. A more generalrelationship among the invariant γ ( V ), the 2nd Frobenius-Schur indicator ν ( V ) andthe pivotal dimension qdim ℓ ( V ) also appears in the case of finite-dimensional pivotalHopf algebras. Definition 4.4. A pivotal element of a Hopf algebra H with the antipode S isgroup-like element g such that S ( h ) = ghg − for all h ∈ H . A Hopf algebra whichadmits a pivotal element is called pivotal . Recall that there are two natural maps, the evaluation map ev : V ∨ ⊗ V → k andthe dual basis or coevaluation map db : k → V ⊗ V ∨ , associated with each finite-dimensional module V over a Hopf algebra H , where { v i } is a basis for V and { v i } is its dual basis in V ∨ .Now, we assume H is a pivotal Hopf algebra with a pivotal element g and antipode S . Suppose V ∈ H - mod fin . Then the map j : V → V ∨∨ defined by j ( v )( f ) = f ( gv ) for all v ∈ V and f ∈ V ∗ , is a pivotal structure of H - mod fin . The (left) pivotal (or quantum) dimensionqdim ℓ ( V ) of V is defined as the scalar corresponding to the k -linear map k db −→ V ∨ ⊗ V ∨∨ id ⊗ j − −−−−→ V ∨ ⊗ V ev −→ k . Direct simplification shows thatqdim ℓ ( V ) = χ V ( g − )where χ V is the character of V . The right pivotal dimension qdim r ( V ) can be definedsimilarly, and qdim r ( V ) = χ V ( g ).Let us identify Hom H ( k , V ⊗ V ) with the H -invariant space ( V ⊗ V ) H . Following[NS2], the map E is defined by(4.5) E ( X i u i ⊗ v i ) = X i v i ⊗ g − u i for all X i u i ⊗ v i ∈ ( V ⊗ V ) H , and(4.6) ν ( V ) = Tr( E ) . Note that E = id and Hom H ( k , V ⊗ V ) ∼ = Hom H ( V ∨ , V ) as k -linear spaces. Thus,if V is absolutely simple, thendim( V ⊗ V ) H = (cid:26) V ∼ = V ∨ , . Therefore, ν ( V ) = ± V ∼ = V ∨ , and 0 otherwise. In particular, Tr( E ) is theeigenvalue of E when V is self-dual. If V is absolutely simple and self-dual in H - mod fin , then dim Hom H ( k , V ∨ ⊗ V ∨ ) =1 = dim Hom H ( V, V ∨ ). Let f ∈ Hom H ( V, V ∨ ) be a non-zero element. Then b ( x, y ) := f ( x )( y ) for all x, y ∈ V, defines an H -invariant non-degenerate bilinear form on V . Conversely, if b ′ is anon-zero H -invariant bilinear form on V , then f ′ ( x ) = b ′ ( x, y ) defines a non-zero H -module map from V to V ∨ . Thus, f ′ is a scalar multiple of f and so b ′ is a scalarmultiple of b .Note that b = P i u ∗ i ⊗ v ∗ i ∈ V ∨ ⊗ V ∨ , and the assignment 1 P i u ∗ i ⊗ v ∗ i ∈ V ∨ ⊗ V ∨ defines a non-zero map in Hom H ( k , V ∨ ⊗ V ∨ ). By (4.5) and (4.6), we find ν ( V ) X i u ∗ i ⊗ v ∗ i = X i v ∗ i ⊗ g − u ∗ i . In terms of b , we have the relation(4.7) ν ( V ) b ( x, y ) = b ( y, gx ) for all x, y ∈ V .
These paragraphs have summarized the Frobenius-Schur Theorem for absolutely sim-ple self-dual modules over a finite-dimensional pivotal Hopf algebra (cf. [LM] and[MN]).
Proposition 4.5.
Let H be a finite-dimensional Hopf algebra over k with antipode S . If H admits a pivotal element g , then (4.8) γ ( V ) = ν ( V ) · qdim ℓ ( V ) = ν ( V ) · qdim r ( V ) . for all absolutely simple V ∈ H - mod fin , where ν ( V ) , qdim ℓ ( V ) and qdim r ( V ) arecomputed using the pivotal element g . In particular, ν ( V ) qdim ℓ ( V ) is independentof the choice of the pivotal element g .Proof. Let V be an absolutely simple H -module. If V is not self-dual, then γ ( V ) = ν ( V ) = 0 and so the equalities hold trivially. Assume V is self-dual, and let { v , . . . , v n } be a basis for V . Then hv j = P nj =1 h ij v i for some h ij ∈ k , and wewrite h for the matrix [ h ij ]. The assignment π : h h defines a matrix represen-tation of H afforded by V with ker π = ann V . Since V is absolutely simple, π issurjective, and hence π induces an algebra isomorphism φ : H/ ann V → M n ( k ) suchthat φ ( h + ann V ) = h = π ( h ) for all h ∈ H .Following the notation in the proof of Proposition 4.3, we let S = φ ◦ S V ◦ φ − .Then S ( X ) = U X t U − , and so S ( X ) = ( U t U − ) − XU t U − for some U ∈ GL n ( k ). On the other hand, S ( X ) = gXg − . Therefore, U t U − g is in the center of M n ( k ) and hence U t U − g = cI for some c ∈ k . Thus, by Lemma 4.2, γ ( V ) = tr( U t U − ) = c tr( g − ) = cχ V ( g − ) . N THE TRACE OF THE ANTIPODE AND HIGHER INDICATORS 17
We claim that c = ν ( V ). Let { e , . . . , e n } denote the standard basis for k n and let( · , · ) denote the standard non-degenerate symmetric bilinear form on k n . We definethe bilinear form b on V by extending the assignment b ( v i , v j ) = ( e i , U − e j ) linearly.Then for h ∈ H , b ( h v i , h v j ) = ( h e i , U − h e j ) = ( e i , h t U − h e j ) = ( e i , U − S ( h ) h e j )= ( e i , U − S ( h ) h e j ) = ε ( h )( e i , U − e j ) = ε ( h ) b ( v i , v j ) . Therefore, b is a non-zero H -invariant form on V . Moreover, b ( v j , gv i ) = ( e j , U − ge i ) = ( e j , ( U − ) t U t U − ge i )= c ( e j , ( U − ) t e i ) = c ( U − e j , e i ) = c b ( v i , v j ) . It follows from (4.7) that c = ν ( V ).Let { u i } be the basis for V such that b ( u i , v j ) = δ ij . Thenqdim r ( V ) = χ V ( g ) = X i b ( gu i , v i ) = X i b ( u i , g − v i ) = χ V ( g − ) = qdim ℓ ( V ) . The third equality is a consequence of the H -invariance of b .The left hand side of the first equation of (4.8) indicates the expression on theright hand side is independent of the choice of the pivotal element g of H . (cid:3) Remark 4.6.
For the quantum group u q ( sl ) at the primitive n th root of unity q with n odd, Jedwab has shown in [J] that γ ( V ) = ( − dim V +1 qdim ℓ V for everysimple module V of dimension less than n . This result together with Proposition 4.5implies that ν ( V ) = ( − dim V +1 for dim V < n . On the values of the indicators
In Section 3, we have seen that the value of ν ( T ) for T a Taft algebra over C is acomplex number. However, if H is a semisimple Hopf algebra over C with antipode S , then ν ( H ) = Tr( S ) is always an integer. This observation follows immediately bythe Larson-Radford theorem [LR], S = id H which implies that all of the eigenvaluesof S are ± Positivity of
Tr( S ) for abelian extensions of Hopf algebras. For a groupalgebra H = C G , it is easy to see that Tr( S ) is equal to the number i G of involutions in G . Therefore, ν ( H ) = i G ≥
1. More generally, we have the following observation.
Proposition 5.1.
Let H = C G C F be a bismash product determined by the factor-izable group L = F G.
Then
Tr( S H ) = Tr( S C L ) = i L . In particular, if H = D ( C G ) ,then Tr( S H ) = i G . Proof.
The first part is [JM, Lemma 2.8]. In the case of D ( C G ), L = G ⋊ G , with G acting on itself by conjugation, and it is easy to see that i L = i G . This is noted in[GM]. (cid:3)
From this proposition, one might hope that if H is semisimple over C , then Tr( S )is always positive. However, this is not the case, even for group-theoretical Hopfalgebras. Example 5.2.
Let H = C G τσ C F where G = h x i×h y i and F = h t i are multiplicativegroups isomorphic to Z × Z and Z respectively, and the coaction ρ and the cocycle σ : F × F → C G are trivial. Let { p g | g ∈ G } denote the dual basis of G for C G . The F -action ⇀ on G , and dual cocycle τ : F → C G ⊗ C G are defined via t ⇀ g = g − for all g ∈ G, and τ ( t ) = X g,h ∈ G τ t ( g, h ) p g ⊗ p h where τ t ( x i y j , x k y l ) = ( − ik + jl + jk . One can verify directly that τ t : G × G → C × isa 2-cocycle on G , that is, τ t satisfies the functional equation τ t ( a, b ) τ t ( ab, c ) = τ t ( a, bc ) τ t ( b, c ) for all a, b, c ∈ G. which makes τ a dual cocycle (see [AD, Proposition 2.16] or [KMM, Lemma 4.5] forthe particular case of cocentral abelian extensions). Moreover, the equality τ t ( a, b ) τ t ( t ⇀ a, t ⇀ b ) = 1 = τ ( a, b )holds for all a, b ∈ G and therefore comultiplication in H is multiplicative. Then itfollows from [AD, Theorem 2.20] that H is a Hopf algebra. The antipode S of H isgiven by S ( p g z ) = τ z ( g − , g ) − p z − ⇀g − z − = τ z ( g − , g ) p z⇀g − z . for z ∈ F and g ∈ G . Note that z ⇀ g − = g if, and only if, z = t or g = 1.Therefore, Tr( S ) = X g ∈ G τ t ( g, g − ) + X g ∈ Gg =1 τ ( g, g − ) . It is easy to see that x , y , x y and 1 are all of the involutions of G , and thus P g ∈ Gg =1 τ ( g, g − ) = 4. Therefore,Tr( S ) = X g ∈ G τ t ( g, g − ) + 4 = X i,j =0 τ t ( x i y j , x − i y − j ) + 4= X i,j =0 ( − − i − j − ij + 4 = 4 −
12 + 4 = − − i + j + ij = (cid:26) i and j are both even, − (cid:3) It is still interesting to know just when Tr( S ) is positive. We give some criteria fora cocentral abelian extensions to have Tr( S ) > Proposition 5.3.
Let G , F be finite groups and H = C G τσ C F a cocentral abelianextension with antipode S . (i) If | F | is odd, then Tr( S ) = i G . N THE TRACE OF THE ANTIPODE AND HIGHER INDICATORS 19 (ii)
Let I F denote the set of all involution in F , F g the stabilizer of g and f F g = { w ∈ F | w ⇀ g = g ± } . Assume that the cocycle σ is trivial, the set I F is a subgroup of F , and F g = f F g for all g ∈ G . Then Tr( S ) > .Proof. Recall that Tr( S ) = X ( g,w ) ∈ G × Fw =1 , w⇀g = g − σ − g ( w, w ) τ − w ( g, g − ) . (i) If | F | is odd, then we haveTr( S ) = X g ∈ G, g = g − σ − g (1 , τ − ( g, g − ) = X g ∈ G, g =1 i G . (ii) Let g ∈ G . Define µ g : C F g → C via µ g ( w ) = τ w ( g, g ) for w ∈ F g . Then by[KMM, Lemma 4.5], since σ is trivial and f F g = F g , µ g is a one-dimensional characterof C F g . ThenTr( S ) = X ( g,w ) ∈ G × Fw =1 , w⇀g = g − τ − w ( g, g − ) = X ( g,w ) ∈ G × Fw =1 , g =1 , w⇀g = g τ − w ( g, g ) = X g ∈ I G X w ∈ I F ∩ F g µ − g ( w ) . By the orthogonality of group characters, we find X w ∈ I F ∩ F g µ − g ( w ) = δ | I F ∩ F g | where δ = 1 if µ g | I F ∩ F g = 1, and δ = 0 otherwise. Therefore, Tr( S ) > (cid:3) Remark 5.4. (i) If F is abelian then I F is a subgroup of F .(ii) If F is cyclic then we may assume that the cocycle σ is trivial (since the group H ( F, ( C G ) × ) is trivial).(iii) In particular, if F is cyclic and G is an elementary abelian 2-group then theconditions of Proposition 5.3 are satisfied and therefore Tr( S ) > Positivity of the indicators of modular quasi-Hopf algebras.
A semisim-ple quasi-triangular quasi-Hopf algebra H over C is said to be modular if the braidedspherical fusion category H - mod fin is a modular tensor category (cf. [ENO] and[BK]). By [M¨u], D ( H )- mod fin is modular for any semisimple quasi-Hopf algebra H over C , where D ( H ) denotes the quantum double (or Drinfeld double) of H . Notethat every semisimple factorizable Hopf algebra H over C is modular (cf. [Ta]).The n -th Frobenius-Schur indicator ν n ( V ) of an object V in a spherical fusioncategory C over C is defined in [NS2, p 71]. In addition, if C is modular, somecanonical linear combination of indicators are always real and non-negative. Proposition 5.5.
Let C be a modular tensor category over C , and U , . . . , U ℓ acomplete list of non-isomorphic simple objects of C with U = I , the unit object.Then X i d i ν n ( U i ) ≥ for all positive integers n , where d i and ν n ( U i ) respectively denote the pivotal dimen-sion and the n -th Frobenius-Schur indicator of U i .Proof. Let θ be the ribbon structure of the modular category C . By [NS1, Theorem7.5], ν n ( U k ) = 1dim C X i,j N kij d i d j ω ni ω nj where N kij = dim C ( U k , U i ⊗ U j ) and ω i ∈ C is given by ω i id U i = θ U i . In particular, ω i is an n -th root of unity where n is the order of θ . Note that P k N kij d k = d i d j , and d k ∈ R for all k by [ENO]. We have X k d k ν n ( U k ) = 1dim C X i,j,k N kij d i d j d k ω ni ω nj = 1dim C X i,j d i d j ω ni ω nj = 1dim C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i d i ω ni (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ , where in the last equation we have used that ω i = ω − i . (cid:3) Theorem 5.6.
Let H be a modular quasi-Hopf algebra over C . Then for all positiveintegers n , the n -th Frobenius-Schur indicator ν n ( H ) is real and non-negative. Inaddition, (i) if H = D ( A ) for some semisimple quasi-Hopf algebra A over C then ν n ( H ) = | ν n ( A ) | ≥ . (ii) If H is a semisimple factorizable Hopf algebra over C with antipode S H then Tr( S H ) ≥ . (iii) If H = D ( A ) for some semisimple Hopf algebra A then Tr( S H ) = Tr( S A ) .Proof. Let U , . . . , U ℓ be a complete list of non-isomorphic simple objects of C = H - mod fin , where U is the trivial H -module C . Then H ∼ = L ℓi =0 d i U i where d i =dim U i . By the additivity of indicators, we have ν n ( H ) = ℓ X i =0 d i ν n ( U i ) . It follows from the proof of Proposition 5.5 that(5.1) ν n ( H ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ dim H ℓ X i =0 d i ω ni (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ n ∈ N , where ω i is the scalar of the ribbon structure component θ U i . N THE TRACE OF THE ANTIPODE AND HIGHER INDICATORS 21 If H = D ( A ) for some semisimple quasi-Hopf algebra A over C , then by [NS1,Theorem 4.1], ν n ( A ) = 1dim A Tr( θ nD ( A ) ⊗ A A ) = 1dim A ℓ X i =0 d i ω ni . It follows from (5.1) that ν n ( H ) = | ν n ( A ) | . If H is a semisimple factorizable Hopf algebra over C , then, by [LM, Theorem 3.1],Tr( S H ) = ν ( H ) ≥ . Moreover, if H = D ( A ) for some semisimple Hopf algebra A , then Tr( S H ) and Tr( S A )are integers, and so we haveTr( S H ) = ν ( H ) = | ν ( A ) | = | Tr( S A ) | = Tr( S A ) . (cid:3) We note that when A = C G and H = D ( A ), we already saw in Proposition 5.1that Tr( S H ) = i G = Tr( S A ) . In general, we can ask the following question.
Question:
For any finite-dimensional complex Hopf algebra H , is it true that ν n ( D ( H )) = | ν n ( H ) | for all positive integer n ?5.3. Prime divisors of the dimension of a semisimple quasi-Hopf algebra.
It is well-known that a group G has even order if, and only if, i G = 1. If S denotesthe antipode of the group algebra H = C G , then dim C G is even if, and only if, ν ( H ) = Tr( S ) = i G = 1. This observation holds for every semisimple Hopf algebraover C . Proposition 5.7.
Let H be a semisimple Hopf algebra over C . Then ν ( H ) = 1 if,and only if, dim H is even.Proof. It is proved in [KSZ1] that dim H is odd if, and only if, the trivial H -module V is the only self-dual simple H -module. Therefore, if dim H is odd, V is the onlysimple H -module V with ν ( V ) = 0. By [LM], we have(5.2) Tr( S ) = ν ( H ) = X V simple dim V · ν ( V )which implies ν ( H ) = dim( V ) · ν ( V ) = 1. Conversely, assume dim( H ) is even. Asnoted above, any eigenvalue of S is 1 or -1. Suppose a of them are 1 and b of themare −
1. Then a + b = dim H is even, so a and b have the same parity and therefore ν ( H ) = Tr( S ) = a − b is also even. Thus ν ( H ) = 1. (cid:3) Using some recent results of [NS1], the preceding proposition can be generalizedto any prime number p for any semisimple quasi-Hopf algebra over C . Theorem 5.8.
Let H be a finite-dimensional semisimple quasi-Hopf algebra over C and p a prime number. Then the following statements are equivalent: (i) p | dim H . (ii) ν p ( V ) = 0 for some non-trivial simple H -module V . (iii) ν p ( H ) = 1 .Proof. By [M¨u], the quantum (Drinfeld) double D ( H ) of H is modular. Let θ bethe ribbon structure on D ( H ) associated with the canonical pivotal structure, N theorder of θ , and ζ N a primitive N -th root of unity.(ii) ⇒ (i): If p ∤ dim H , then p ∤ N (cf. [NS1, Theorems 8.4 or 9.1] or [Et]).Therefore, σ : ζ N ζ pN defines an automorphism of Q ( ζ N ) / Q . Moreover, by [NS1,Theorem 4.1],(5.3) ν p ( V ) = 1dim H Tr( θ pK ( V ) )for V ∈ H - mod fin , where K ( V ) = D ( H ) ⊗ H V . Let ˆΓ = { U , . . . , U ℓ } be a completeset of non-isomorphic simple D ( H )-modules, and ω i be the scalar of the component θ U i . Then ω i is a power of ζ N , and so σ ( ω i ) = ω pi . Now let V ∈ H - mod fin be simple.One can consider the action of the Galois group on the indicators (cf. [KSZ2, p 24]or [NS1, p 62]), ν p ( V ) = 1dim H Tr( θ pK ( V ) ) = 1dim H ℓ X i =0 N i d i ω pi = σ H ℓ X i =0 N i d i ω i ! = σ ( ν ( V ))where N i = dim Hom D ( H ) ( K ( V ) , U i ), d i = dim U i . Note that ν ( V ) is equal to 0 if V is not the unit object and 1 otherwise. Thus, ν p ( V ) = 0 for all non-trivial simple H -modules V .(iii) ⇒ (ii): Let V , V , . . . , V n form a complete set of non-isomorphic simple H -modules with V being the trivial H -module. Since H = L ni =0 (dim V i ) V i , ν p ( H ) =1 + P ni =1 (dim V i ) ν p ( V i ). Thus, ν p ( H ) = 1 implies that ν p ( V i ) = 0 for some i > ⇒ (iii): Suppose ν p ( H ) = 1. Recall from [NS2, p 71] that ν p ( H ) is the ordinarytrace of a C -linear automorphism E on Hom H ( C , H ⊗ p ) and E p = id. Therefore, bya linear algebra argument (cf. [KSZ2, p 26]), ν p ( H ) = Tr( E ) ≡ dim Hom H ( C , H ⊗ p ) mod p . Since dim Hom H ( C , H ⊗ p ) = (dim H ) p − , we have1 ≡ (dim H ) p − mod p Therefore, p ∤ dim H . (cid:3) Remark 5.9.
In the case of semisimple Hopf algebras H , the indicator was definedas ν n ( V ) = χ V (Λ [ n ] ) . In that case N = exp( H ) and exp( H ) | (dim H ) [EG1] . Itwas shown in [KSZ2] that there exists a linear operator E on Hom H ( C , V ⊗ n ) suchthat E n = id , Tr( E ) = ν n ( V ) , and Equation 5.3 holds, where now θ ± is the Drinfeldelement of D ( H ) . Then Theorem 5.8 can be proved in the context of semisimpleHopf algebras with the same arguments as above, replacing some facts on quasi-Hopfalgebras with corresponding results established in [KSZ2] and [EG1] . N THE TRACE OF THE ANTIPODE AND HIGHER INDICATORS 23
6. Tr( S ) and the degrees of representations According to [Is, p 54] and [JL, p 278], one application of Frobenius-Schur indica-tors for finite groups is to give an easier proof of the Brauer-Fowler theorem. Thistheorem states that for a given positive integer n , there exist only finitely many sim-ple groups G containing an involution x such that the centralizer of x has order n;this was useful in the classification of finite simple groups. Thus it seems worthwhileto try to find Hopf algebra analogs of these methods.The main preliminary step in both [Is] and [JL] shows that H has a non-trivialrepresentation whose degree is bounded by a function involving Tr( S ). This stepgeneralizes to Hopf algebras, provided we replace i G in the statement for C G byTr( S ) in our result for H . Theorem 6.1.
Let H be a semisimple Hopf algebra of even dimension n over C andlet S be its antipode. Then there exists an irreducible character χ = χ ∗ = ε such that deg( χ ) ≤ α := n − | Tr( S ) − | . Moreover if
Tr( S ) > then ν ( χ ) = 1 , and if Tr( S ) < then ν ( χ ) = − .Proof. The formula is obviously well-defined since Tr( S ) = 1 by Proposition 5.7. Alsonote that n = dim H = 1 + P χ = ε χ (1 H ) , where as before χ runs through the set Irr( H )of irreducible characters of H . Let T = { χ ∈ Irr( H ) | ν ( χ ) = 1 and χ = ε } , and T ′ = { χ ∈ Irr( H ) | ν ( χ ) = − } . Obviously, ε / ∈ T ′ since ν ( ε ) = 1. From (5.2), Tr( S ) = P χ ν ( χ ) χ (1 H ). Therefore,since ν ( χ ) ∈ { , , − } ,(6.1) Tr( S ) − X χ ∈T χ (1 H ) − X χ ∈T ′ χ (1 H ) . Since Tr( S ) = 1, there are two possibilities: either Tr( S ) > S ) <
1. Firstassume that Tr( S ) >
1. Then by Equation (6.1)0 < Tr( S ) − ≤ X χ ∈T χ (1 H )and, in particular, T is not empty. Thus, by the Cauchy-Schwartz inequality, wehave (Tr( S ) − ≤ ( X χ ∈T χ (1 H )) ≤ |T | X χ ∈T ( χ (1 H )) ≤ |T | ( n − X χ ∈T ( χ (1 H )) ≤ X χ = ε ( χ (1 H )) = n − ≤ |T | ( n − (Tr( S ) − . Therefore there exists χ ∈ T such that χ (1 H ) ≤ n − S ) − The case that Tr( S ) < < − Tr( S ) ≤ X χ ∈T ′ χ (1 H )and, in particular, T ′ is non-empty. By the same argument, we have X χ ∈T ′ ( χ (1 H )) ≤ n − ≤ |T ′ | ( n − (1 − Tr( S )) . which implies χ (1 H ) ≤ n − − Tr( S ) . (cid:3) Remark 6.2.
In view of Theorem 5.8, ν ( H ) = 1 for any semisimple quasi-Hopfalgebra H of even dimension. By the same proof, Theorem 6.1 remains to hold forany even dimensional semisimple quasi-Hopf algebra over C if one replaces Tr( S ) by ν ( H ) . Acknowledgement:
The third author would like to thank Susan Montgomery andthe University of Southern California for their hospitality during his sabbatical leave.
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