Central invariants and higher indicators for semisimple quasi-Hopf algebras
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- by Siu-Hung Ng and Peter Schauenburg PDF
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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.References
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Additional Information
- Siu-Hung Ng
- Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
- MR Author ID: 343929
- Email: rng@iastate.edu
- Peter Schauenburg
- Affiliation: Mathematisches Institut der Universität München, Theresienstr. 39, 80333 München, Germany
- MR Author ID: 346687
- Email: schauenburg@math.lmu.de
- Received by editor(s): October 11, 2005
- Published electronically: 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 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 1839-1860
- MSC (2000): Primary 16W30, 18D10, 81R05
- DOI: https://doi.org/10.1090/S0002-9947-07-04276-6
- MathSciNet review: 2366965