A Maschke type theorem for relative Hom-Hopf modules

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.

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Maschke-type theorem and Morita context over weak Hopf algebras

Science in China Series A Mathematics ◽

10.1007/s11425-006-0587-6 ◽

2006 ◽

Vol 49 (5) ◽

pp. 587-598 ◽

Cited By ~ 15

Author(s):

Liangyun Zhang

Keyword(s):

Hopf Algebras ◽

Type Theorem ◽

Morita Context ◽

Maschke Type Theorem ◽

Weak Hopf Algebras

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SMASH PRODUCTS, SEPARABLE EXTENSIONS AND A MORITA CONTEXT OVER HOPF ALGEBROIDS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813501247 ◽

2014 ◽

Vol 13 (04) ◽

pp. 1350124

Author(s):

YONG WANG ◽

GUANGQUAN GUO

Keyword(s):

Type Theorem ◽

Smash Product ◽

Module Algebra ◽

Morita Context ◽

Hopf Algebroid ◽

Hopf Algebroids ◽

Maschke Type Theorem ◽

Module Algebras ◽

Product Algebra ◽

Separable Extensions

Let [Formula: see text] be a Hopf algebroid, and A a left [Formula: see text]-module algebra. This paper is concerned with the smash product algebra A#H over Hopf algebroids. In this paper, we investigate separable extensions for module algebras over Hopf algebroids. As an application, we obtain a Maschke-type theorem for A#H-modules over Hopf algebroids, which generalizes the corresponding result given by Cohen and Fischman in [Hopf algebra actions, J. Algebra100 (1986) 363–379]. Furthermore, based on the work of Kadison and Szlachányi in [Bialgebroid actions on depth two extensions and duality, Adv. Math.179 (2003) 75–121], we construct a Morita context connecting A#H and [Formula: see text] the invariant subalgebra of [Formula: see text] on A.

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SEPARABLE FUNCTORS FOR THE CATEGORY OF PARTIAL ENTWINED MODULES

Journal of Algebra and Its Applications ◽

10.1142/s0219498812501010 ◽

2012 ◽

Vol 11 (05) ◽

pp. 1250101

Author(s):

SHUANG-JIAN GUO ◽

SHUAN-HONG WANG

Keyword(s):

Type Theorem ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Maschke Type Theorem ◽

Necessary And Sufficient

We give necessary and sufficient conditions for the functor F from the category of partial entwined modules to the category of right A-modules to be separable. This leads to a generalized notion of integrals of partial entwining structures. As an application, we prove a Maschke-type theorem for partial entwined modules.

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MASCHKE-TYPE THEOREM AND DUALITY THEOREM FOR WEAK TWISTED SMASH PRODUCTS

Taiwanese Journal of Mathematics ◽

10.11650/twjm/1500406492 ◽

2011 ◽

Vol 15 (6) ◽

pp. 2701-2719

Author(s):

Xiao-yan Zhou

Keyword(s):

Duality Theorem ◽

Type Theorem ◽

Maschke Type Theorem

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

Advances in Mathematical Physics ◽

10.1155/2021/6677332 ◽

2021 ◽

Vol 2021 ◽

pp. 1-10

Author(s):

Bingliang Shen ◽

Ling Liu

Keyword(s):

Hopf Algebra ◽

Type Theorem ◽

Smash Product ◽

Morita Context ◽

Maschke Type Theorem

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 .

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Galois corings and groupoids acting partially on algebras

Journal of Algebra and Its Applications ◽

10.1142/s021949882140003x ◽

2020 ◽

pp. 2140003

Author(s):

S. Caenepeel ◽

T. Fieremans

Keyword(s):

Galois Extension ◽

Galois Theory ◽

Type Theorem ◽

Partial Action ◽

Morita Theory ◽

Maschke Type Theorem ◽

Galois Corings ◽

Partial Galois Extension ◽

Special Case

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.

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