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

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Let H be a finite dimensional quasi-Hopf algebra over a field k and A a right H-comodule algebra in the sense of Hausser and Nill. We first show that on the k-vector space A H* we can define an algebra structure, denoted by A H*, in the monoidal category of left H-modules (i.e. A H* is an H-module algebra. Then we will prove that the category of two-sided (A, H)-bimodules is isomorphic to the category of relative (A H*, H*)-Hopf modules, as introduced in by Hausser and Nill. In the particular case where 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.