The relative modular object and Frobenius extensions of finite Hopf algebras

Abstract: For a certain kind of tensor functor $F: \mathcal{C} \to \mathcal{D}$, we define the relative modular object $\chi_F \in \mathcal{D}$ as the "difference" between a left adjoint and a right adjoint of $F$. Our main result claims that, if $\mathcal{C}$ and $\mathcal{D}$ are finite tensor categories, then $\chi_F$ can be written in terms of a categorical analogue of the modular function on a Hopf algebra. Applying this result to the restriction functor associated to an extension $A/B$ of finite-dimensional Hopf algebras, we recover the result of Fischman, Montgomery and Schneider on the Frobenius type property of $A/B$. We also apply our results to obtain a "braided" version and a "bosonization" version of the result of Fischman et al.