Central Invariants and Higher Indicators for Semisimple Quasi-hopf Algebras

In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules of a semisimple quasi-Hopf algebra H via the categorical counterpart developed in a 2005 preprint. When H is an ordinary Hopf algebra, we show that our definition coincides with that introduced by Kashina, Sommerhäuser, and Zhu. We find a sequence of gauge invariant central elements of H such that the higher FS-indicators of a module V are obtained by applying its character to these elements. As an application, we show that FS-indicators are sufficient to distinguish the four gauge equivalence classes of semisimple quasi-Hopf algebras of dimension eight corresponding to the four fusion categories with certain fusion rules classified by Tambara and Yamagami. Three of these categories correspond to well-known Hopf algebras, and we explicitly construct a quasi-Hopf algebra corresponding to the fourth one using the Kac algebra. We also derive explicit formulae for FS-indicators for some quasi-Hopf algebras associated to group cocycles. Introduction The (degree 2) Frobenius-Schur indicator ν2(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. In particular, the FrobeniusSchur indicator of a simple module is non-zero if, and only if, the simple module is self-dual, and its value can only be 0, 1 or −1. In proving that ±1 are the only possible non-zero values for the Frobenius-Schur indicator of a simple module of a semisimple quasi-Hopf algebra H over C [MN05], the fact that H-modfin is pivotal, proved by Etingof, Nikshych, and Ostrik [ENO], has been used. Based on the pivotal structure, the second author [Sch04] later introduced a categorical definition of degree 2 Frobenius-Schur indicators and gave a different proof of the Frobenius-Schur Theorem for quasi-Hopf algebras. The higher indicators of irreducible representations of a finite group do not have a direct interpretation as the degree 2 indicators (cf. [Isa94]). The n-th FrobeniusSchur indicator of a finite-dimensional module V with character χ of a semisimple Received by the editors October 11, 2005. 2000 Mathematics Subject Classification. Primary 16W30, 18D10, 81R05. The first author was supported by the NSA grant number H98230-05-1-0020. The second author was supported by a DFG Heisenberg fellowship. c ©2007 American Mathematical Society