Relative Serre functor for comodule algebras

TL;DR

This paper explicitly describes the relative Serre functor for module categories over Hopf algebras and comodule algebras, linking it to Frobenius structures and pivotal structures, with concrete examples.

Contribution

It provides an explicit description of the relative Serre functor in the setting of Hopf algebra modules and comodule algebras, connecting it to Frobenius structures.

Findings

Explicit formula for the relative Serre functor in this setting

Connection between Serre functor and Frobenius structures

Analysis of pivotal structures and concrete examples

Abstract

Let $\mathcal{C}$ be a finite tensor category, and let $\mathcal{M}$ be an exact left $\mathcal{C}$-module category. The relative Serre functor of $\mathcal{M}$ is an endofunctor $\mathbb{S}$ on $\mathcal{M}$ together with a natural isomorphism $\underline{\mathrm{Hom}}(M, N)^* \cong \underline{\mathrm{Hom}}(N, \mathbb{S}(M))$ for $M, N \in \mathcal{M}$, where $\underline{\mathrm{Hom}}$ is the internal Hom functor of $\mathcal{M}$. In this paper, we discuss the case where $\mathcal{C}$ and $\mathcal{M}$ are the category of modules over a finite-dimensional Hopf algebra $H$ and the category of modules over an $H$-comodule algebra $L$, respectively. We give an explicit description of the relative Serre functor of $\mathcal{M}$ and its twisted module structure in terms of the Frobenius structure of $L$. We also study pivotal structures on $\mathcal{M}$ and give some concrete examples.