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

In this paper, we obtain a canonical central element v(H) for each semi-simple quasi-Hopf algebra H over any field k and prove that v(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 chi, chi(v(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 chi(v(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 chi(v(H)), and certain associated bilinear forms when the underlying field k is algebraically closed of characteristic zero. (C) 2003 Elsevier Inc. All rights reserved.