Central Invariants and Higher Indicators for Semisimple Quasi-Hopf Algebras · arXivDesk
math/0508140Aug 8, 2005The higher Frobenius-Schur indicators for certain quasi-Hopf algebras associated with finite groups and their 3-cocycles have been computed in section 7
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 V of a semisimple quasi-Hopf algebra H via the categorical counterpart developed in NS05. We prove that this definition of higher FS-indicators coincides with the higher indicators introduced by Kashina, Sommerhäuser, and Zhu when H is a Hopf algebra. We also obtain a sequence of canonical central elements of H, which is invariant under gauge transformations, whose values, when evaluated by the character of an H-module V
Nearby in the stack
, are the higher Frobenius-Schur indicators of
V
. 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. We also derive explicit formulae for FS-indicators for some quasi-Hopf algebras associated to group cocycles.