Crossed modules for Lie-Rinehart algebras.
type: Item
instance of: scholarly article
title: Crossed modules for Lie-Rinehart algebras.
zbMATH Open document ID: 1046.17006
author: M. Ladra
author: J. M. Casas
author: Teimuraz Pirashvili
published in: Journal of Algebra
publication date: 2004-05-27T00:00:00Z
review text: Let \(k\) be a field and \(A\) be a commutative \(k\)-algebra. A {Lie-Rinehart algebra} over \(A\) is a Lie \(k\)-algebra \(\mathcal{L}\) which also has the structure of an \(A\)-module. Further \(\mathcal{L}\) must admit an appropriately compatible Lie \(k\)-algebra and an \(A\)-module homomorphism to the set of \(k\)-derivations on \(A\). Note that any Lie \(A\)-algebra can be considered as a Lie-Rinehart algebra over \(A\) in a natural way. The authors introduce the notion of a {crossed module} of Lie-Rinehart algebras over \(A\). A crossed module consists of a pair \((\mathcal{L},\mathcal{R})\) where \(\mathcal{L}\) is a Lie-Rinehart algebra over \(A\) and \(\mathcal{R}\) is a Lie \(A\)-algebra. Further, \(\mathcal{L}\) is assumed to act on \(\mathcal{R}\) (i.e., there exists a map \(\mathcal{L}\otimes\mathcal{R} \to \mathcal{R}\) satisfying certain properties) and there exists a Lie \(k\)-algebra homomorphism \(\mathcal{R} \to \mathcal{L}\) which is well-behaved with respect to the actions of \(A\) and \(\mathcal{L}\). This definition extends the definition of crossed modules for a Lie algebra introduced by \textit{C. Kassel} and \textit{J.-L. Loday} [Ann. Inst. Fourier 32, 119--142 (1982; Zbl 0485.17006)]. The main goal of this paper is to generalize the work of Kassel and Loday to show that these newly defined crossed modules can be classified by the third cohomology of Lie-Rinehart algebras.
reviewed by: Christopher P. Bendel
Mathematics Subject Classification ID: 17B55
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17B56
zbMATH DE Number: 2066828
zbMATH Keywords: Lie-Rinehart algebras
zbMATH Keywords: crossed modules
zbMATH Keywords: cohomology of Lie algebras
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.10.001
OpenAlex ID: W1984496949
Wikidata QID: Q115351793
cites work: Q5817473
cites work: Crossed \(n\)-fold extensions of groups and cohomology
cites work: Poisson cohomology and quantization.
cites work: Central extensions of Lie algebras
cites work: Cohomologie et groupe de Steinberg rélatifs
cites work: Differential Forms on General Commutative Algebras
DOI: 10.1016/J.JALGEBRA.2003.10.001
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description: scientific article; zbMATH DE number 2066828