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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) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

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