The five-dimensional complete left-symmetric algebra structures compatible with an abelian Lie algebra structure
type: Item
instance of: scholarly article
title: The five-dimensional complete left-symmetric algebra structures compatible with an abelian Lie algebra structure
zbMATH Open document ID: 0891.17024
DOI: 10.1016/S0024-3795(96)00591-5
author: Karel Dekimpe
author: Paul Igodt
author: Veerle Ongenae
published in: Linear Algebra and its Applications
publication date: 1998-08-02T00:00:00Z
review text: A left symmetric algebra (LSA) \({\mathcal L}\) is a vector space with a bilinear multiplication satisfying the following identity \[ X(YZ)- (XY)Z= Y(XZ)- (YX)Z \quad \forall X,Y,Z \in {\mathcal L}. \] The LSA's considered in this paper are also commutative, i.e. \(XY=YX\) for all \(X, Y\in {\mathcal L}\). The calculation \(X(ZY)= X(YZ)= Y(XZ)= (XZ)Y\) shows that the commutative LSA are precisely the commutative associate algebras. An LSA \({\mathcal L}\) is called complete if the map \({\mathcal L}\to {\mathcal L}\), \(X\mapsto X+XY\) is an isomorphism for all \(Y\in {\mathcal L}\). One can show that under these conditions the left multiplications by elements of \({\mathcal L}\) are all nilpotent. The goal of the paper is to classify the five-dimensional real commutative LSA's. If, in a commutative associate algebra, the left multiplication \(\lambda_Y: X\mapsto YX\) is nilpotent, say \(\lambda^m_Y=0\), then \((1+\lambda_Y) \sum^{m-1}_{k=0} (-1)^k \lambda^k_Y =1\) so that the above classification problem amounts to classifying the five-dimensional commutative abelian associative real algebras in which all left multiplications are nilpotent. The authors obtain their classification by describing central extensions of LSA's in a cohomological language largely parallel to the corresponding theories for groups or Lie algebras.
Mathematics Subject Classification ID: 17D99
zbMATH DE Number: 1067958
zbMATH Keywords: left symmetric algebra
zbMATH Keywords: five-dimensional real commutative LSA
zbMATH Keywords: five dimensional commutative abelian associative real algebras
zbMATH Keywords: central extensions
reviewed by: Joachim Hilgert
MaRDI profile type: Publication
Wikidata QID: Q115340000
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description: scientific article; zbMATH DE number 1067958