Higher Frobenius-schur Indicators for Pivotal Categories

We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a k-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a k-linear semisimple pivotal monoidal category — where both notions are defined —, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms. Introduction The classical (degree two) Frobenius-Schur indicator ν2(V ) of an irreducible representation V of a finite group G has been generalized to Frobenius-Schur indicators of simple modules of semisimple Hopf algebras by Linchenko and Montgomery [12], to certainC-fusion categories by Fuchs, Ganchev, Szlachányi, and Vescernyés [5], and further to simple objects in pivotal (or sovereign) categories by Fuchs and Schweigert [6]; Mason and Ng [13] treated the case of simple modules over semisimple quasi-Hopf algebras. As in the classical case, the indicator always takes one of the values 0,±1, and is related to the question if and how the representation in consideration is self-dual. In the last case, proving that ν2(V ) ∈ {0,±1} uses the result of Etingof, Nikshych, and Ostrik [4] that the module category of a semisimple complex quasi-Hopf algebra is a pivotal monoidal category. A different proof based on this pivotal structure and the description of indicators in [5] was given in [17]. The higher indicators νn(V ) of an irreducible group representation V , which have less obvious meaning for the structure of V , were generalized to simple modules of a semisimple Hopf algebra by Kashina, Sommerhäuser, and Zhu [10]. In the present paper we define and study higher Frobenius-Schur indicators νn(V ) for an object V of a k-linear pivotal monoidal category C. We do this with a view towards the categories of modules over semisimple complex quasi-Hopf algebras, though we will only give the (quite involved) explicit formulas and examples for that case in another paper [14]. Our definition of νn(V ) is as the trace of an endomorphism E (n) V of the vector space of morphisms C(I, V ), where I is the unit object. The endomorphism arises from a special case of a map C(I, V ⊗W ) → C(I,W ⊗V ) defined for any two objects in terms of duality and the pivotal structure. In the case where V is an H-module for a semisimple Hopf algebra H , we can identify C(I, V ) with the invariant subspace (V ) H , and E (n) V is given by a cyclic permutation; the description of νn(V ) as a trace in this case is contained in [10]. If n = 2 and V is simple, then C(I, V ⊗ V ) is one-dimensional or vanishes. Thus E (2) V is (at most) a scalar, which coincides with its trace; that computing the trace in a different way leads to the indicator formulas from [13] was shown in [17]. An endomorphism of C(V , V ) conjugate to E (2) V is also used to describe the degree two indicator in [5] (and to define E (2) V in [17]). The maps E (n) V have been studied in connection with 3-manifold invariants by Gelfand and Kazhdan [7]. We prove that the higher indicators are invariant under equivalences of pivotal monoidal categories, and that equivalences of pseudo-unital fusion categories (which are pivotal categories by [4]) are automatically pivotal equivalences. While these invariance properties are to be expected 1 2 SIU-HUNG NG AND PETER SCHAUENBURG from the categorical nature of the definitions, the proofs are not obvious. In particular, it would be tautological to assume to be dealing with strict categories (on grounds that every monoidal category is equivalent to a strict one) to prove invariance of certain properties under equivalence. Barrett and Westbury [2] have proved (although starting from a different set of axioms) that every pivotal monoidal category is equivalent to a strict one. By the invariance results we have proved, it is then sufficient to assume such a simplified structure of C when proving general properties of E (n) V and νn(V ). We prove, in section 5, that the order of E (n) V divides n; this is stated in [7] and proved for n = 2 (and C strict). In particular, the possible values of the higher Frobenius-Schur indicators are cyclotomic integers; this is well-known for the group case, where the higher indicators are in fact always integers, and proved for the Hopf algebra case in [10]. Finally, we show that the FrobeniusSchur indicator of an object V in a pivotal fusion category can be computed as the trace (in a suitable sense) of a natural endomorphism of V , which we call the Frobenius-Schur endomorphism. The organization of the paper is as follows: we cover in Section 1 some basic definitions, notations, conventions and preliminary results of pivotal monoidal categories for the remaining discussion. In Section 2, we give a proof that every pivotal monoidal category is equivalent, as pivotal monoidal categories, to a strict one. We then define a sequence of scalars νn(V ), called the higher FrobeniusSchur indicators, for each object V in a k-linear pivotal monoidal category in Section 3. We prove in Section 4 that these higher Frobenius-Schur indicators are invariant under k-linear pivotal monoidal equivalences. This result allows one to study these indicators by considering only the strict k-linear pivotal monoidal categories and we prove in Section 5 that all the higher Frobenius-Schur indicators for a k-linear pivotal monoidal category are cyclotomic integers provided the characteristic of the algebraically closed field k is zero. In Section 6, we consider the higher Frobenius-Schur indicators for a pseudo-unitary fusion category over C. In this case, we show that these indicators are invariant under k-linearly monoidal equivalence. Finally, in Section 7, we show that the nth Frobenius-Schur indicator of an object V in a semisimple k-linear pivotal monoidal category isthe pivotal trace of a natural endomorphism FS (n) V , called the Frobenius-Schur endomorphism. In another paper [14] we will study the pivotal fusion categories of modules over a semisimple complex quasi-Hopf algebra. In this case the Frobenius-Schur endomorphisms correspond to central gauge invariants in the quasi-Hopf algebras. The indicators are obtained by applying the representation’s character, which corresponds precisely to the definition of higher indicators in [10] for the case of ordinary Hopf algebras. Acknowledgements. The first author acknowledges support from the NSA grant number H9823005-1-0020. The second author thanks the Deutsche Forschungsgemeinschaft for support by a Heisenberg fellowship, the Institute of Mathematics of Tsukuba University for its hospitality, and Akira Masuoka for being the perfect host. 1. Preliminaries We will first fix some conventions: In a monoidal category C, the associativity isomorphism is Φ: (U ⊗ V )⊗W → U ⊗ (V ⊗W ). We will make the assumption that the unit object in a monoidal category is always strict, V ⊗ I = V = I ⊗V . As pointed out in [16], this assumption can always be made true after replacing the tensor product in C with an isomorphic one. For the purpose of this paper however (where frequently the question of invariance of certain constructions under tensor equivalences is key) this may not be a rigorous justification; we simply make the assumption that I is a strict unit for simplicity, and hold that the general case can be treated by an insignificant but annoying expansion of all proofs. By the well-known coherence theorem for monoidal categories, if X,Y ∈ C are formed by tensoring the same sequence of objects V1, . . . , Vn ∈ C, only with different placement of parentheses, then there is a unique morphism Φ : X → Y composed formally from HIGHER FROBENIUS-SCHUR INDICATORS FOR PIVOTAL CATEGORIES 3 instances of Φ and Φ. (By a “formal” composition of instances of Φ and Φ we mean one that could be written down in a suitably defined free category, excluding compositions that only become possible because “formally different” objects happen to be identical in the concrete category at hand; such accidental composites may of course fail to agree.) As a simple example Φ = Φ(T ⊗ Φ) = Φ(Φ ⊗W )Φ : T ⊗ ((U ⊗ V )⊗W ) → (T ⊗ U)⊗ (V ⊗W ). A monoidal functor (F , ξ) : C → D will have the structure isomorphism ξ : F(V ) ⊗ F(W ) → F(V ⊗ W ). We will assume that the structure isomorphism for the unit objects is the identity F(I) = I. Given a general monoidal functor (F , ξ, ξ0) with ξ0 : I → F(I) not the identity, this can always be achieved by replacing F with an isomorphic functor F ′ whose object map is given by F (X) = F(X) if X 6= I, and F (I) = I. Let X be obtained from tensoring a sequence V1, . . . , Vn of objects of C with some choice of parentheses, and let X ′ be obtained from V ′ i = F(Vi) in the same way. We will use the following special case of coherence of monoidal functors: There is a unique formal composition ξ : X ′ → F(X) of instances of ξ, and if Y, Y ′ are obtained from the same sequence of objects with a different placement of parentheses, then X ′ Φ ξ // F(X) F(Φ) Y ′ ξ // F(Y ) commutes. The coherence result of Epstein [3] treats the case of symmetric monoidal functors between symmetric monoidal categories; the diagram above is contained in the appropriate analog for the non-symmetric case, obtained essentially by just leaving out the symmetry. This is certainly folklore; a proof can be extracted from [3]. A (left) dual object of V ∈ C is a triple (V , ev, db) with an object V ∨ ∈ C and morphisms ev : V ∨ ⊗ V → I and db: I → V ⊗ V ∨ such that the compositions V db⊗V −−−−→ (V ⊗ V )⊗ V Φ −→ V ⊗ (V ∨ ⊗ V ) V ⊗ev −−−−→ V, V ∨ V ⊗db −−−−−→ V ∨ ⊗ (V ⊗ V ) Φ −−−→ (V ∨ ⊗ V )⊗ V ∨ ev⊗V ∨ −−−−−→ V ∨ are identities. We say that C is (left) rigid if every object has a dual. We use the notation V for the symmetric notion of a right dual of an object V ∈ C; this is the same as a left dual in the category C in which the tensor product is defined in the reverse order. For any object V of a monoidal category having a dual object V , we obtain an adjunction A0 : C(U, V ⊗W ) ∼= C(V ∨ ⊗ U,W ) by A0(f) = ( V ∨ ⊗ U V ⊗f −−−−→ V ∨ ⊗ (V ⊗W ) Φ −−−→ (V ∨ ⊗ V )⊗W ev⊗W −−−−→ W )