Two-sided (two-cosided) Hopf modules and Doi-Hopf modules for quasi-Hopf algebras

Let $H$ be a finite dimensional quasi-Hopf algebra over a field $k$ and $\mathfrak{A}$ a right $H$-comodule algebra in the sense of Hausser and Nill. We first show that on the $k$-vector space $\mathfrak{A}\text{ot}{H}^{\ast }$ we can define an algebra structure, denoted by $\mathfrak{A}\text{ovsm}{H}^{\ast }$, in the monoidal category of left $H$-modules (i.e. $\mathfrak{A}\text{ovsm}{H}^{\ast }$ is an $H$-module algebra. Then we will prove that the category of two-sided $({\mathfrak A}, H)$-bimodules$\hba $isisomorphictothecategoryofrelative$({\mathfrak A}\ovsm H^, H^)$-Hopf modules, as introduced in by Hausser and Nill. In the particular case where $\mathfrak{A}=H$, we will obtain a result announced by Nill. We will also introduce the categories of Doi-Hopf modules and two-sided two-cosided Hopf modules and we will show that they are in certain situations isomorphic to module categories.