Embeddability of Mal'tsev algebras in Lie algebras
type: Item
instance of: scholarly article
title: Embeddability of Mal'tsev algebras in Lie algebras
zbMATH Open document ID: 0798.17020
DOI: 10.1007/BF02259849
published in: Algebra and Logic
publication date: 1994-04-12T00:00:00Z
full work available at URL: https://eudml.org/doc/187605
review text: Let \(L\) be a Lie algebra over a field of characteristic \(\ne 2,3\), and let \(\sigma\), \(\rho\) be nonidentity automorphisms of \(L\) such that \(\sigma^2= \text{Id}\), \(\rho^3= \text{Id}\), \(\sigma\rho= \rho^2\sigma\). Then \(H= \{x\in L\mid \sigma(x)=x\}\) is a subalgebra of \(L\) and \(M= \{x\in L\mid \sigma(x) =-x\}\) is a subspace. Suppose that \((\rho^2+ \rho+ \text{Id})(x)=0\) for \(x\in M\). Then \(L=M+ \rho(H)\) (vector space direct sum). Let \(\pi: L\to M\) be the projection with kernel \(\rho(H)\). If multiplication is defined on \(M\) by \(x\cdot y= \pi([ x,y ])\), where \([x,y]\) is the multiplication in \(L\), then \(M(\cdot)\) is a Mal'tsev algebra. Conversely, given any Mal'tsev algebra \(M(\cdot)\) over a field of characteristic \(\ne 2,3\), then there is a Lie algebra \(L\) which satisfies the indicated conditions and has \(M(\cdot)\) as the Mal'tsev algebra just constructed.
Mathematics Subject Classification ID: 17D10
zbMATH DE Number: 536658
zbMATH Keywords: embeddability Lie algebras
zbMATH Keywords: Mal'cev algebra
author: Pavel O. Mikheev
MaRDI profile type: Publication
Wikidata QID: Q115392646
cites work: Q5513077
cites work: Über eine Beziehung zwischen Malcev-Algebren und Lie-Tripelsystemen
cites work: Simple Moufang loops
OpenAlex ID: W2053723915
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description: scientific article; zbMATH DE number 536658