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 right H-comodule algebra. We introduce the category of two-sided Hopf modules, and prove that it is isomorphic to a module category. We also show that two-sided Hopf modules are coalgebra over a certain comonad. We introduce Doi–Hopf modules, and show that they are comodules over a certain coring. If the underlying H-module coalgebra is finite-dimensional, then Doi–Hopf modules are modules over a certain smash products. A similar result holds for two-sided two-cosided Hopf modules.