The five-dimensional complete left-symmetric algebra structures compatible with an abelian Lie algebra structure
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.
five dimensional commutative abelian associative real algebras
scientific article; zbMATH DE number 1067958