Frobenius-Schur indicator for categories with duality

TL;DR

This paper introduces a categorical Frobenius-Schur indicator to unify and generalize various Frobenius-Schur theorems across different algebraic structures, providing new insights and applications.

Contribution

It develops a category-theoretical framework for Frobenius-Schur indicators, including twisted versions, applicable to quasi-Hopf algebras and quantum groups.

Findings

Unified Frobenius-Schur indicator framework for categories with duality

Derived a twisted Frobenius-Schur theorem for semisimple quasi-Hopf algebras

Applied the theory to quantum SL_2 and related structures

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 how the `twisted' theory arises from the ordinary case. As a demonstration, we give a twisted Frobenius-Schur theorem for semisimple quasi-Hopf algebras. We also give several applications to the quantum SL_2.