TY - JOUR

T1 - Relative Serre functor for comodule algebras

AU - Shimizu, Kenichi

N1 - Publisher Copyright:
Copyright © 2019, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2019/3/31

Y1 - 2019/3/31

N2 - 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.

AB - 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.

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