Frobenius-Schur theorem for $C^{\ast }$-categories

Abstract

We generalize the Frobenius-Schur theorem to $C^{\ast }$-categories. From this category-theoretical point of view, we introduce the notions of real, complex and quaternionic representations of Hopf $C^{\ast }$-algebras. Based on these definitions, we give another type of the Hopf-algebraic analogue of the Frobenius-Schur theorem, originally due to Linchenko and Montgomery. Also given are similar results for weak Hopf $C^{\ast }$-algebras, table algebras and compact quantum groups.

arXiv comment34 pages. Completely rewritten. The title has been changed since the main result is reformulated in a $C^{\ast }$-categorical setting

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