Central Invariants and Frobenius-Schur Indicators for Semisimple Quasi-Hopf Algebras

Abstract

In this paper, we obtain a canonical central element ${\nu }_H$ for each semi-simple quasi-Hopf algebra $H$ over any field $k$ and prove that ${\nu }_H$ 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 $\chi$, $\chi ({\nu }_H)$ 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 $\chi ({\nu }_H)$ is the corresponding Frobenius-Schur Indicator. We also prove an analog of a Theorem of Larson-Radford for split semi-simple quasi-Hopf algebra over any field $k$. Using this result, we establish the relationship between the antipode $S$, the values of $\chi ({\nu }_H)$, and certain associated bilinear forms when the underlying field $k$ is algebraically closed of characteristic zero.

arXiv comment32 pages (version 3)

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• On the trace of the antipode and higher indicators
• Gauge invariants from the powers of antipodes
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• Indicators of Tambara-Yamagami categories and Gauss sums
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• The Witt group of a braided Monoidal category
• A Morita context and Galois extensions for Quasi-Hopf algebras
• A trace-like invariant for representations of Hopf algebras
• Cocycle deformations and Galois objects of semisimple Hopf algebras of dimension $16$
• On indicators of Hopf algebras
• Remarks on the McKay Conjecture
• Higher Indicators for the Doubles of some Totally Orthogonal Groups
• Frobenius-Schur theorem for $C^{\ast }$-categories
• Involutory quasi-Hopf algebras
• Representations of some Hopf algebras associated to the symmetric group S_n