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 algebras 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.