Frobenius and Mashke type Theorems for Doi-Hopf modules and entwined modules revisited: a unified approach

Frobenius properties and Maschke-type theorems for entwined modules

Entwined modules arose from the coalgebra-Galois theory. They are a generalisation of unified Doi-Hopf modules. In this paper, Frobenius properties and Maschke-type theorems, known for Doi-Hopf modules are extended to the case of entwined modules.

Dual Entwining Structures and Dual Entwined Modules

In this note we introduce and investigate the concepts of dual entwining structures and dual entwined modules. This generalizes the concepts of dual Doi-Koppinen structures and dual Doi-Koppinen modules introduced (in the infinite case over rings) by the author is his dissertation.

缠绕余模与缠绕反模范畴的非交换基变换的Galois测量

Galois measurings for noncommutative base change of entwined contramodule and entwined comodule categories

本文基于两个模似范畴研究entwining structure的非交换基变换，探讨由measurings诱导的范畴间映射表现为Galois扩张的准则，并给出此类扩张具备可分性、Frobenius或Maschke型性质的条件。

Hom-entwining structures and Hom-Hopf-type modules

The notions of Hom-coring, Hom-entwining structure and associated entwined Hom-module are introduced. A theorem regarding base ring extension of a Hom-coring is proven and then is used to acquire the Hom-version of Sweedler coring. Motivated by the work of T. Brzezinski, a Hom-coring associated to an entwining Hom-structure is constructed and an identification of entwined Hom-modules with Hom-comodules of this Hom-coring is shown. The dual algebra of this Hom-coring is proven to be a $ψ$-twisted convolution algebra. By a construction, it is shown that a Hom-Doi-Koppinen datum comes from a Hom-…

Yetter-Drinfeld modules over weak Hopf algebras and the center construction

We introduce Yetter-Drinfeld modules over a weak Hopf algebra $H$, and show that the category of Yetter-Drinfeld modules is isomorphic to the center of the category of $H$-modules. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. If $H$ is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show …

Maschke functors, semisimple functors and separable functors of the second kind. Applications

We introduce separable functors of the second kind (or $H$-separable functors) and $H$-Maschke functors. $H$-separable functors are generalizations of separable functors. Various necessary and sufficient conditions for a functor to be $H$-separable or $H$-Maschke, in terms of generalized (co)Casimir elements (integrals, in the case of Hopf algebras), are given. An $H$-separable functor is always $H$-Maschke, but the converse holds in particular situations. A special role will be played by Frobenius functors and their relations to $H$-separability. Our concepts are applied to modules, comodules…