The Category of Partial Doi-Hopf Modules and Functors

Let (H, A, C), (H', A', C') be two partial Doi-Hopf datums consisting of a Hopf algebra H, a partial right H-comodule algebra A and a partial right H-module coalgebra. Given alpha : H -> H', beta: A -> A' and gamma: C -> C', we define an induction functor between the category M(H)(A)(C) of all partial Doi-Hopf modules and the category M(H')(A')(C'), and we prove that this functor has a right adjoint. Specially, we then give necessary and sufficient conditions for the functor F: M(H)(A)(C) -> M(H)(A) (exactly the category of right A-modules). This leads to a generalized notion of integrals. Moreover, from these results, we deduce a version of Maschke-type Theorems for partial Doi-Hopf modules. The applications of our results are considered.