Central invariants and Frobenius-Schur indicators for semisimple quasi-Hopf algebras |
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Authors: | Geoffrey Mason Siu-Hung Ng |
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Affiliation: | a Mathematics Department, University of California, Santa Cruz, CA 95064, USA b Department of Mathematics, Iowa State University, Ames, IA 50011-2064, USA c Mathematics Department, Towson University, Baltimore, MD 21252, USA |
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Abstract: | In this paper, we obtain a canonical central element νH for each semi-simple quasi-Hopf algebra H over any field k and prove that νH is invariant under gauge transformations. We show that if k is algebraically closed of characteristic zero then for any irreducible representation of H which affords the character takes only the values 0, 1 or −1, moreover if H is a Hopf algebra or a twisted quantum double of a finite group then χ(νH) is the corresponding Frobenius-Schur indicator. We also prove an analog of a theorem of Larson-Radford for split semi-simple quasi-Hopf algebras over any field k. Using this result, we establish the relationship between the antipode S, the values of χ(νH), and certain associated bilinear forms when the underlying field k is algebraically closed of characteristic zero. |
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Keywords: | Frobenius-Schur indicators Gauge equivalence Tensor categories Quasi-Hopf algebras |
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