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

Abstract

We study the following question: when is the right adjoint of the forgetful functor from the category of ( H , A , C ) (H,A,C) -Doi-Hopf modules to the category of A A -modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that C ⊗ A C\otimes A and the smash product A # C ∗ A\# C^* are isomorphic as ( A , A # C ∗ ) (A, A\# C^*) -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 A = H A=H , and this leads to the notion of k k -Frobenius H H -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 H H is finite dimensional and unimodular.