The Maschke-Type Theorem and Morita Context for BiHom-Smash Products

Let H , α H , β H , ω H , ψ H , S H be a BiHom-Hopf algebra and A , α A , β A be an H , α H , β H -module BiHom-algebra. Then, in this paper, we study some properties on the BiHom-smash product A # H . We construct the Maschke-type theorem for the BiHom-smash product A # H and form an associated Morita context A H , A H A A # H , A # H A A H , A # H .

Galois corings and groupoids acting partially on algebras

Bagio and Paques [Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra 40 (2012) 3658–3678] developed a Galois theory for unital partial actions by finite groupoids. The aim of this note is to show that this is actually a special case of the Galois theory for corings, as introduced by Brzeziński [The structure of corings, Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Represent. Theory 5 (2002) 389–410]. To this end, we associate a coring to a unital partial action of a finite groupoid on an algebra [Formula: see text], and show that this coring is Galois if and only if [Formula: see text] is an [Formula: see text]-partial Galois extension of its coinvariants.

Entwined Hom-modules and frobenius properties

Entwined Hom-modules were introduced by Karacuha in [13], which can be viewed as a generalization of Doi-Hom Hopf modules and entwined modules. In this paper, the sufficient and necessary conditions for the forgetful functor F : ?H(Mk)(?)CA ? ?H(Mk)A and its adjoint G : ?H(Mk)A ? ?H (Mk)(?)CA form a Frobenius pair are obtained, one is that A?C and the C*?A are isomorphic as (A;C*op#A)-bimodules, where (A,C,?) is a Hom-entwining structure. Then we can describe the isomorphism by using a generalized type of integral. As an application, a Maschke type theorem for entwined Hom-modules is given.

An Application of Rafael's Theorem over Partial Entwined Modules

We obtain necessary and sufficient conditions for the functor [Formula: see text] on the category of partial entwined modules that forgets the A-action to be separable. As an application, we prove a Maschke-type theorem for the category of partial entwined modules.