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

Abstract: Frobenius-Schur indicators 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 partially dualized quasi-Hopf algebra; 2) We show that the property in Kashina's conjecture on the exponent holds for a class of semisimple Hopf algebras fitting into abelian extensions with assumptions weaker than split.