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Axioms 2012, 1(3), 324-364; doi:10.3390/axioms1030324

Frobenius–Schur Indicator for Categories with Duality

Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan
Received: 2 July 2012 / Revised: 23 August 2012 / Accepted: 29 September 2012 / Published: 23 October 2012
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)
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We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we establish twisted versions of the Frobenius–Schur theorem for various algebraic objects. We also give several applications to the quantum SL2.
Keywords: Frobenius–Schur indicator; category with duality; Hopf algebra; quantum groups Frobenius–Schur indicator; category with duality; Hopf algebra; quantum groups
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Shimizu, K. Frobenius–Schur Indicator for Categories with Duality. Axioms 2012, 1, 324-364.

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