Entwined Hom-modules and Frobenius properties

Abstract

Entwined Hom-modules were introduced by Karacuha, which as a generalization of Doi-Hom Hopf modules and entwined modules. In this paper, the sufficient and necessary conditions for the forgetful functor
$F: \widetilde{\mathscr{H}}(\mathscr{M}_k)(\psi)^{C}_{A} \rightarrow \widetilde{\mathscr{H}}(\mathscr{M}_k)_{A}$ and its
adjoint $G: \widetilde{\mathscr{H}}(\mathscr{M}_k)_{A} \rightarrow \widetilde{\mathscr{H}}(\mathscr{M}_k)(\psi)^{C}_{A} $ form a Frobenius pair are obtained, 
one is that $A\o C$ and the $C^{\ast}\o A$ are isomorphic as $(A; C^{\ast op}\#A)$-bimodules,
where $(A, C, \psi)$ is a Hom-entwining structure. Then we can describe the isomorphism by using a generalized type of integral. As an application, a Maschke type theorem for entwined Hom-modules is given.