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. © 2007 American Mathematical Society.
Publication Source (Journal or Book title)
Transactions of the American Mathematical Society
Ng, S., & Schauenburg, P. (2008). Central invariants and higher indicators for semisimple quasi-Hopf algebras. Transactions of the American Mathematical Society, 360 (4), 1839-1860. https://doi.org/10.1090/S0002-9947-07-04276-6