Relative Lie algebra cohomology
type: Item
instance of: scholarly article
title: Relative Lie algebra cohomology
zbMATH Open document ID: 0682.17009
DOI: 10.1016/0022-4049(89)90053-4
author: Mark Muzere
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let \((\mathfrak g,\mathfrak f)\) be a pair of finite dimensional Lie algebras where \(\mathfrak f\) is a Lie subalgebra of \(\mathfrak g\), and let \((U\mathfrak g,U\mathfrak f)\) be the corresponding pair of universal enveloping algebras. The author studies the Hochschild relative cohomology groups and the Hochschild complex [see \textit{G. Hochschild}, Trans. Am. Math. Soc. 82, 246--269 (1956; Zbl 0070.26903)]. It is known that if the Hochschild complex has a U\({\mathfrak f}\)-homotopy then the relative cohomology groups \(H^*({\mathfrak g},{\mathfrak f},V)\) and \(H^*(U{\mathfrak g},U{\mathfrak f},V)\) are identical. The purpose of the paper is to investigate the existence of a U\({\mathfrak f}\)-homotopy and the possible isomorphism of the two relative cohomology groups. It is shown that the relative cohomology groups are related by a spectral sequence and some good conditions are given for the isomorphism. A vanishing property for the Hochschild relative cohomology groups is obtained and it is shown that in general the two cohomology groups are not isomorphic.
reviewed by: Alice Fialowski
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B55
zbMATH DE Number: 4116773
zbMATH Keywords: finite dimensional Lie algebra
zbMATH Keywords: Hochschild relative cohomology groups
zbMATH Keywords: Hochschild complex
zbMATH Keywords: \(U\mathfrak f\)-homotopy
zbMATH Keywords: vanishing property
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90053-4
OpenAlex ID: W2086770372
Wikidata QID: Q115364443
cites work: Q4705361
cites work: Q3240881
cites work: Cohomology Theory of Lie Groups and Lie Algebras
cites work: Relative Homological Algebra and Homological Dimension of Lie Algebras
cites work: Foundations of relative homological algebra
cites work: Q5648522
cites work: Relative Homological Algebra
cites work: Cohomology of Lie algebras
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Q5343374
cites work: The cohomology of restricted Lie algebras and of Hopf algebras
cites work: Q5341863
cites work: RELATIVE HOMOLOGICAL ALGEBRA IN CATEGORIES OF MODULES
Recommended article: Leibniz and Hochschild homology
Recommended article: Q4693951
Recommended article: Invariant syzygies and semisimple groups
Recommended article: A complex for the cohomology of restricted Lie algebras
Recommended article: Universal algebra of a Hom-Lie algebra and group-like elements
Recommended article: Formes harmoniques et cohomologie relative des algèbres de Lie
Recommended article: Relative Lie Algebra Cohomology Revisited
Recommended article: Do $n$-Lie algebras have universal enveloping algebras
Recommended article: Cyclic homology of universal enveloping algebras of Lie algebras of dimension 3
Recommended article: The first Hochschild cohomology as a Lie algebra
description: scientific article; zbMATH DE number 4116773