Frobenius-Schur theorem for $C^*$-categories

TL;DR

This paper extends the Frobenius-Schur theorem to $C^*$-categories, introducing new notions of representations for Hopf $C^*$-algebras and providing algebraic analogues for various quantum algebraic structures.

Contribution

It generalizes the Frobenius-Schur theorem within a category-theoretic framework and defines real, complex, and quaternionic representations for Hopf $C^*$-algebras.

Findings

New category-theoretic Frobenius-Schur theorem for $C^*$-categories

Definitions of real, complex, and quaternionic representations for Hopf $C^*$-algebras

Analogues of the theorem for weak Hopf $C^*$-algebras, table algebras, and compact quantum groups

Abstract

We generalize the Frobenius-Schur theorem to $C^*$-categories. From this category-theoretical point of view, we introduce the notions of real, complex and quaternionic representations of Hopf $C^*$-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 $C^*$-algebras, table algebras and compact quantum groups.