Strong connections and invertible weak entwining structures.
type: Item
instance of: scholarly article
title: Strong connections and invertible weak entwining structures.
zbMATH Open document ID: 1267.16031
author: J. M. Fernández Vilaboa
author: Ramón González Rodríguez
author: J. N. Alonso Alvarez
publication date: 2012-11-07T00:00:00Z
review text: Let \(\mathcal C\) be a strict monoidal category. A right-right weak entwining structure in \(\mathcal C\) is a triple \((A,C,\psi_R)\), where \(A\) is an algebra, \(C\) is a coalgebra and \(\psi_R\colon C\otimes A\to A\otimes C\) is a morphism satisfying some compatibility conditions. The category whose objects are \(M\) with structure of right \(A\)-module and right \(C\)-comodule and a compatibility condition is denoted by \(\mathcal M_A^C(\psi_R)\), the category of right-right weak entwined modules. If \((A,C,\psi_R)\) is a right-right weak entwining structure and there exists a coaction \(\rho_A\) such that \((A,\mu_A\rho_A)\in\mathcal M_A^C(\psi_R)\), it is possible to obtain an extension \(A^R_C\to A\) and a canonical morphism. If this morphism is an isomorphism, it is called a right \(C\)-Galois extension. The authors introduce the notion of strong connection for an invertible weak entwining structure. The main result of the paper (Theorem 3.12) establishes that the existence of a strong connection form is equivalent, under certain conditions, to the bijectivity of the canonical morphism of a weak Galois extension associated to a weak invertible entwining structure. An explicit formula of a strong connection for an invertible weak entwining structure with coseparable coalgebra is obtained, generalizing the formula by \textit{E. J. Beggs} and \textit{T. Brzeziński} [Appl. Categ. Struct. 16, No. 1-2, 57-63 (2008; Zbl 1195.16030)].
Mathematics Subject Classification ID: 16T15
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 18D10
zbMATH DE Number: 6102801
zbMATH Keywords: Hopf algebras
zbMATH Keywords: coalgebras
zbMATH Keywords: monoidal categories
zbMATH Keywords: invertible weak entwining structures
zbMATH Keywords: strong connections
zbMATH Keywords: weak Galois extensions
reviewed by: Blas Torrecillas
MaRDI profile type: Publication
full work available at URL: s10114-012-0088-1
OpenAlex ID: W1982638221
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DOI: 10.1007/S10114-012-0088-1
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6102801