Doi-Hopf modules, Yetter-Drinfel'd modules and Frobenius type properties

We study the following question: when is the right adjoint of the forgetful functor from the category of -Doi-Hopf modules to the category of -modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that and the smash product are isomorphic as -bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case , and this leads to the notion of -Frobenius -module coalgebra. In the special case of Yetter-Drinfel'd modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if is finite dimensional and unimodular.