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Central invariants and higher indicators for semisimple quasi-Hopf algebras

Author(s): Siu-Hung Ng; Peter Schauenburg
Journal: Trans. Amer. Math. Soc. 360 (2008), 1839-1860.
MSC (2000): Primary 16W30, 18D10, 81R05
Posted: October 30, 2007
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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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Additional Information:

Siu-Hung Ng
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: rng@iastate.edu

Peter Schauenburg
Affiliation: Mathematisches Institut der Universität München, Theresienstr. 39, 80333 München, Germany
Email: schauenburg@math.lmu.de

DOI: 10.1090/S0002-9947-07-04276-6
PII: S 0002-9947(07)04276-6
Received by editor(s): October 11, 2005
Posted: October 30, 2007
Additional Notes: The first author was supported by the NSA grant number H98230-05-1-0020.
The second author was supported by a DFG Heisenberg fellowship.
Copyright of article: Copyright 2007, American Mathematical Society

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