Free left-symmetric algebras and an analogue of the Poincaré-Birkhoff- Witt theorem
type: Item
instance of: scholarly article
title: Free left-symmetric algebras and an analogue of the Poincaré-Birkhoff- Witt theorem
zbMATH Open document ID: 0831.17001
published in: Journal of Algebra
publication date: 1996-02-20T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 17A50
zbMATH DE Number: 579548
zbMATH Keywords: free algebras
zbMATH Keywords: Lie algebra
zbMATH Keywords: solvable affine groups
author: Dan Segal
reviewed by: Joachim Hilgert
MaRDI profile type: Publication
full work available at URL: jabr.1994.1088
OpenAlex ID: W1986732903
DOI: 10.1006/JABR.1994.1088
description: scientific article; zbMATH DE number 579548