Central Invariants and Higher Indicators for Semisimple Quasi-Hopf Algebras

Details Central Invariants and Higher Indicators for Semisimple Quasi-Hopf Algebras Central Invariants and Higher Indicators for Semisimple Quasi-Hopf Algebras

2005-08-08

arxiv.org/abs/math/0508140v2

Trans. Amer. Math. Soc. 360 (2008), 1839-1860 (Final Version)

Mathematics

Quantum Algebra

The higher Frobenius-Schur indicators for certain quasi-Hopf algebras associated with finite groups and their 3-cocycles have

Scientific paper

10.1090/S0002-9947-07-04276-6

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 \cite{NS05}. We prove that this definition of higher FS-indicators coincides with the higher indicators introduced by Kashina, Sommerh\"auser, 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 $V$, 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.

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