Relative Serre functor for comodule algebras

Abstract

Let $\mathcal{C}$ be a finite tensor category, and let $\mathcal{M}$ be an exact left $\mathcal{C}$-module category. A relative Serre functor of $\mathcal{M}$, introduced by Fuchs, Schaumann and Schweigert, 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 $\mathcal{C} = {}_H \mathfrak{M}$ and $\mathcal{M} = {}_L \mathfrak{M}$ for a finite-dimensional Hopf algebra $H$ and a finite-dimensional exact left $H$-comodule algebra $L$. We give an explicit description of a relative Serre functor of ${}_L \mathfrak{M}$ and its twisted module structure in terms of integrals of $H$ and the Frobenius structure of $L$. We also study pivotal structures on ${}_L \mathfrak{M}$ and give some explicit examples.Comment: 48 page