The Category of Partial Doi–Hopf Modules and Functors

Abstract

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