Central invariants and higher indicators for semisimple quasi-Hopf algebras

Ng, Siu-Hung; Schauenburg, Peter

doi:10.1090/S0002-9947-07-04276-6

Central invariants and higher indicators for semisimple quasi-Hopf algebras
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by Siu-Hung Ng and Peter Schauenburg PDF

Trans. Amer. Math. Soc. 360 (2008), 1839-1860 Request permission

Abstract:

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.

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