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

Abstract

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

Frobenius properties and Maschke-type theorems for entwined modules

“…The aim of this article is to show that results of recent papers [6] and [7] concerning Frobenius properties and a Maschke-type theorem for Doi-Hopf modules [9] [11] hold for the more general class of modules, known as entwined modules [3]. These are modules of an algebra and comodules of a coalgebra such that the action and the coaction satisfy certain compatibility condition.…”

Frobenius and separable functors for the category of entwined modules over cowreaths, II: Applications

“…Theorem 4.5 generalizes [16,Theorem 4.2] to the quasi-Hopf algebra setting. Note that our approach is different from the one in [16]. Proof.…”

“…Frobenius/separable functors are strictly related to Frobenius/separable algebra extensions, see [11,14]. From this perspective, in Hopf algebra theory, the study of Frobenius and separability for Doi-Hopf modules was done in [13,15,16]. Afterwards this study was refined and applied to entwined modules by Brzeziński in [2].…”

Entwined modules over linear categories and Galois extensions

“…For Doi-Hopf modules, Frobenius and separability conditions were studied extensively in a series of papers [13], [14], [15]. Later, Brzeziński studied Frobenius and Maschke type theorems for entwined modules in [4].…”