Lax entwining structures, groupoid algebras and cleft extensions.
type: Item
instance of: scholarly article
title: Lax entwining structures, groupoid algebras and cleft extensions.
zbMATH Open document ID: 1333.16030
author: J. N. Alonso Alvarez
author: J. M. Fernández Vilaboa
author: C. Soneira Calvo
author: Ramón González Rodríguez
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2014-06-24T00:00:00Z
review text: An entwining structure consists of an algebra \(A\) and a coalgebra \(C\) together with a map \(\psi\colon C\otimes A\to A\otimes C\) satisfying appropriate axioms. Examples can be constructed starting from a bialgebra \(H\) (playing the role of \(C\)) and an \(H\)-comodule algebra \(A\). Galois theory for entwining structures can be developed (leading to so-called coalgebra Galois extensions), and this is a generalization of Hopf-Galois theory. Generalizations of entwining structures are possible in different directions: weak entwining structures can be constructed from comodule algebras over weak bialgebras; partial entwining structures can be constructed from partial actions of bialgebras on algebras. These two notions unify in lax entwining structures. The aim of this paper is to develop Galois theory of lax entwining structures. The main results are in Sec. 3, where lax \(C\)-Galois extensions with normal basis are characterized. The authors introduce lax \(C\)-cleft extensions and prove that lax \(C\)-cleft extensions are precisely lax \(C\)-Galois extensions with normal basis (Theorem 3.11). This generalizes a classical result of \textit{Y. Doi} and \textit{M. Takeuchi} [Commun. Algebra 14, 801-817 (1986; Zbl 0589.16011)].
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16T15
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 20L05
zbMATH DE Number: 6307365
zbMATH Keywords: lax entwining structures
zbMATH Keywords: partial entwining structures
zbMATH Keywords: weak entwining structures
zbMATH Keywords: weak Hopf algebras
zbMATH Keywords: groupoid algebras
zbMATH Keywords: Galois objects
zbMATH Keywords: cleft extensions
zbMATH Keywords: normal bases
zbMATH Keywords: coalgebra Galois extensions
zbMATH Keywords: comodule algebras
zbMATH Keywords: partial actions of bialgebras
reviewed by: Stefaan Caenepeel
MaRDI profile type: Publication
full work available at URL: s00574-014-0044-z
OpenAlex ID: W2081287016
cites work: Weak \(C\)-cleft extensions, weak entwining structures and weak Hopf algebras.
cites work: Weak \(C\)-cleft extensions and weak Galois extensions.
cites work: Weak braided Hopf algebras
cites work: Weak Hopf algebras. I: Integral theory and \(C^*\)-structure
cites work: Morita theory for coring extensions and cleft bicomodules.
cites work: Coalgebra extensions and algebra coextensions of galois type
cites work: On modules associated to coalgebra Galois extensions
cites work: Modules over weak entwining structures
cites work: Partial (Co)Actions of Hopf Algebras and Partial Hopf-Galois Theory
cites work: Hopf algebras and Galois theory
cites work: On the Group Ring
cites work: Partial representations and partial group algebras
cites work: Associativity of crossed products by partial actions, enveloping actions and partial representations
cites work: Partial actions and Galois theory
cites work: Q3717232
cites work: Twisted Partial Actions: A Classification of Regular C*-Algebraic Bundles
cites work: A characterization of the cleft comodule triples
cites work: Q3897166
DOI: 10.1007/S00574-014-0044-Z
Recommended article: Invertible Lax Entwining Structures and C-Cleft Extensions
Recommended article: Coalgebra extensions and algebra coextensions of galois type
Recommended article: Group Entwining Structures and Group Coalgebra Galois Extensions
Recommended article: Galois and cleft monoidal cowreaths. Applications
Recommended article: Weak \(C\)-cleft extensions and weak Galois extensions.
Recommended article: Cleft and Galois extensions associated with a weak Hopf quasigroup
Recommended article: Q3507159
Recommended article: Strong connections and invertible weak entwining structures.
Recommended article: Coalgebra-Galois extensions from the extension theory point of view
Recommended article: On Doi-Hopf modules and Yetter-Drinfeld modules in symmetric monoidal categories.
description: scientific article; zbMATH DE number 6307365