Frobenius-Schur Indicators of Dual Fusion Categories and Semisimple Partially Dualized Quasi-Hopf Algebras

TL;DR

This paper extends Frobenius-Schur indicator theory to dual categories of spherical fusion categories, establishing relations between indicators and algebraic structures like Hopf algebras and quasi-Hopf algebras, with applications to exponents and module indicators.

Contribution

It introduces a formula for indicators in dual categories of spherical fusion categories and relates these indicators to properties of semisimple Hopf and quasi-Hopf algebras, including exponents.

Findings

Established relations between indicators of dual categories and original categories.

Derived formulas for indicators of modules over specific semisimple Hopf algebras.

Proved that for semisimple left partially dualized quasi-Hopf algebras, exponent equals Frobenius-Schur exponent.

Abstract

Frobenius-Schur indicators (or indicators for short) of objects in pivotal monoidal categories were defined and formulated by Ng and Schauenburg in 2007. In this paper, we introduce and study an analogous formula for indicators in the dual category $\mathcal{C}_\mathcal{M}^\ast$ to a spherical fusion category $\mathcal{C}$ (with respect to an indecomposable semisimple module category $\mathcal{M}$) over $\mathbb{C}$. Our main theorem is a relation between indicators of specific objects in $\mathcal{C}_\mathcal{M}^\ast$ and $\mathcal{C}$. As consequences: 1) We obtain equalities on the indicators between certain representations and the exponents of a semisimple complex Hopf algebra as well as its left partially dualized quasi-Hopf algebra; 2) We deduce formulas on indicators of certain modules over some particular semisimple Hopf algebras - bismash products and quantum doubles; 3) We show that for each semisimple left partially dualized quasi-Hopf algebra, its exponent and Frobenius-Schur exponent are identical.