Central invariants and Frobenius–Schur indicators for semisimple quasi-Hopf algebras

Abstract

In this paper, we obtain a canonical central element ν H for each semi-simple quasi-Hopf algebra H over any field k and prove that ν H is invariant under gauge transformations. We show that if k is algebraically closed of characteristic zero then for any irreducible representation of H which affords the character χ, χ(ν H ) takes only the values 0, 1 or -1, moreover if H is a Hopf algebra or a twisted quantum double of a finite group then χ(ν H ) is the corresponding Frobenius-Schur Indicator. We also prove an analog of a Theorem of Larson-Radford for split semi-simple quasi-Hopf algebra over any field k. Using this result, we establish the relationship between the antipode S, the values of χ(ν H ), and certain associated bilinear forms when the underlying field k is algebraically closed of characteristic zero.

Higher Frobenius-Schur indicators for pivotal categories

“…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 classical (degree two) Frobenius-Schur indicator ν 2 (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 certain C * -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.…”

Central invariants and higher indicators for semisimple quasi-Hopf algebras

“…The (degree 2) Frobenius-Schur indicator ν 2 (V ) of an irreducible representation V of a finite group G has been generalized to simple modules of semisimple Hopf algebras by Linchenko and Montgomery [LM00], to certain C * -fusion categories by Fuchs, Ganchev, Szlachányi, and Vecsernyés [FGSV99], and to simple modules of semisimple quasi-Hopf algebras by Mason and the first author [MN05]. A more general version of the Frobenius-Schur Theorem holds for the simple modules of semisimple Hopf algebras or even quasi-Hopf algebras.…”

Frobenius–Schur indicators for a class of fusion categories

“…Let G be a finite group and let ω be 3-cocycle on G. Consider the Dijkgraaf-Pasquier-Roche quasi-Hopf algebra D ω G, also called the twisted quantum double of G [Dijkgraaf et al 1991]. By the results in [Natale 2003], a semisimple quasi-Hopf algebra H is group-theoretical if and only if its quantum double is gauge equivalent to a quasi-Hopf algebra D ω G. The FrobeniusSchur indicators for D ω G have been computed in [Mason and Ng 2005], and seen to coincide in this case with the indicators introduced by Bantay [1997].…”

“…In Sections 2 and 3 we recall the definition of the indicators constructed in [Mason and Ng 2005] and the definition and main properties of group-theoretical categories as given in [Ostrik 2002;. In Section 4 we give a description, up to gauge equivalence, of the structure of group-theoretical quasi-Hopf algebras, and finally in Section 5 we give an explicit formula for the Frobenius-Schur indicators of group-theoretical categories.…”