On Total Frobenius-Schur Indicators

TL;DR

This paper introduces the total Frobenius-Schur indicator for objects in spherical fusion categories, proving its invariance and non-negativity, and explores its implications for classifying quasi-Hopf algebras versus Hopf algebras.

Contribution

It defines the total Frobenius-Schur indicator, proves its invariance and integrality properties, and establishes positivity as a criterion for gauge equivalence to Hopf algebras.

Findings

Total indicators are invariants of spherical fusion categories.

Total indicators are non-negative integers for representation categories of semisimple quasi-Hopf algebras.

Positivity of total indicators is necessary for a quasi-Hopf algebra to be gauge equivalent to a Hopf algebra.

Abstract

We define total Frobenius-Schur indicator for each object in a spherical fusion category $C$ as a certain canonical sum of its higher indicators. The total indicators are invariants of spherical fusion categories. If $C$ is the representation category of a semisimple quasi-Hopf algebra $H$, we prove that the total indicators are non-negative integers which satisfy a certain divisibility condition. In addition, if $H$ is a Hopf algebra, then all the total indicators are positive. Consequently, the positivity of total indicators is a necessary condition for a quasi-Hopf algebra being gauge equivalent to a Hopf algebra. Certain twisted quantum doubles of finite groups and some examples of Tambara-Yamagami categories are discussed for the sufficiency of this positivity condition.