Free left-symmetric algebras and an analogue of the Poincaré-Birkhoff- Witt theorem
A not necessarily associative algebra \(A\) over a ring \(k\) is called left symmetric if it satisfies the identity \(a(bc)- (ab)c= b(ac)- (ba)c\). If \(A\) is left symmetric, the usual commutator product \([a,b ]= ab- ba\) satisfies the Jacobi identity, so that \(A\) carries the structure of a Lie algebra. The author defines a free left symmetric algebra and describes an explicit basis for this algebra. Moreover he introduces a universal enveloping left-symmetric algebra for a given Lie algebra \(L\) and proves an analogue of the Poincaré-Birkhoff-Witt theorem in the case that \(L\) is a free \(k\)-module. In an appendix the author recalls how left- symmetric algebras occur in the study of simply transitive group actions of solvable affine groups.
scientific article; zbMATH DE number 579548