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

Article

Frobenius–Schur Indicator for Categories with Duality

Kenichi Shimizu

Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan

Received: 2 July 2012; in revised form: 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)

Download PDF Full-Text [443 KB, uploaded 23 October 2012 08:57 CEST]

Abstract: 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

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