Relative Serre functor for comodule algebras

Let C be a finite tensor category, and let M be an exact left C-module category. The relative Serre functor of M, introduced by Fuchs, Schaumann and Schweigert, is an endofunctor S on M together with a natural isomorphism Hom(M, N)^∗ ≅ Hom(N, S(M)) for M, N ∈ M, where Hom is the internal Hom functor of M. In this paper, we discuss the case where C = _HM and M = _LM for a finite-dimensional Hopf algebra H and a finite-dimensional exact left H-comodule algebra L. We give an explicit description of the relative Serre functor of _LM and its twisted module structure in terms of integrals of H and the Frobenius structure of L. We also study pivotal structures on _LM and give some explicit examples.