Leibniz homology of extended Lie algebras
type: Item
instance of: scholarly article
title: Leibniz homology of extended Lie algebras
zbMATH Open document ID: 0894.17002
DOI: 10.1023/A:1007778622807
author: Allahtan Victor Gnedbaye
published in: \(K\)-Theory
publication date: 1998-06-02T00:00:00Z
review text: Let \(A\) be a commutative algebra, \({\mathfrak g}\) be a Lie algebra, and consider the extended Lie algebra \(A\otimes{\mathfrak g}\) with bracket \([a_1\otimes x_1,a_2\otimes x_2]= a_1a_2\otimes [x_1,x_2]\). For \({\mathfrak g}\) perfect the Leibniz homology \(HL_* \otimes{\mathfrak g}\) in dimensions two and three are computed in terms of the Hochschild homology of \(A\) and the coinvariants of the symmetric powers of \({\mathfrak g}\). Leibniz homology is a noncommutative analogue of Lie algebra homology which was introduced by Loday six years ago. The results here parallel those of \textit{A. Haddi} [Homologie de degré trois des algèbre de Lie étendues à une algèbre commutative, C. R. Acad. Sci., Paris, Sér. I 321, 965-968 (1995; Zbl 0846.17024)] obtained for the usual Lie algebra cohomology of \(A\otimes{\mathfrak g}\) where the Hochschild homology of \(A\) is replaced by its cyclic homology.
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 17B55
zbMATH DE Number: 1132930
zbMATH Keywords: extended Lie algebra
zbMATH Keywords: Leibniz homology
zbMATH Keywords: Hochschild homology
Wikidata QID: Q123253480
reviewed by: U. Tillmann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1132930