Central Invariants and Frobenius-Schur Indicators for Semisimple Quasi-Hopf Algebras

Abstract

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