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 A a right H-comodule algebra in the sense of [12]. 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 Hmodule algebra in the sense of [2]). Then we will prove that the category of two-sided (A,H)bimodules HM H A is isomorphic to the category of relative (A # H ∗ ,H)-Hopf modules, as introduced in [3]. In the particular case where A = H, we will obtain the Nill’s result announced in [14]. 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.