The relative modular object and Frobenius extensions of finite Hopf algebras

TL;DR

This paper introduces the relative modular object for tensor functors between finite tensor categories, linking it to Hopf algebra modular functions, and applies it to Frobenius extensions of finite-dimensional Hopf algebras, recovering and extending known results.

Contribution

It defines the relative modular object in categorical terms and relates it to Hopf algebra modular functions, providing new insights into Frobenius extensions and their braided and bosonization variants.

Findings

The relative modular object can be expressed via categorical analogues of Hopf algebra modular functions.

The framework recovers the Frobenius property for finite-dimensional Hopf algebra extensions.

Extensions to braided and bosonization contexts are established.

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.