The Category of Partial Doi-Hopf Modules and Functors

Abstract

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 ${\alpha}: H \rightarrow H '$, ${\beta}: A \rightarrow A '$ and ${\gamma}: C \rightarrow C'$, we define an induction functor between the category ${\cal M}(H)^{C}_{A}$ of all partial Doi-Hopf modules and the category ${\cal M}(H')^{C'}_{A'}$, and we prove that this functor has a right adjoint. Specially, we then give necessary and sufficient conditions for the functor $F\kern -1pt :\kern -1pt {\cal M}(H)^{C}_{A} \kern -1pt \rightarrow \kern -1pt {\cal M}(H)_{A}$ (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.