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

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.