Lie Algebras and Cotriangular Spaces

Abstract

Let $\Pi=(P,L)$ be a partial linear space in which any line contains three points and let $K$ be a field. Then by ${\cal L}_K(\Pi)$ we denote the free $K$-algebra generated by the elements of $P$ and subject to the relations $xy=0$ if $x$ and $y$ are noncollinear elements from $P$ and $xy=z$ for any triple $\{x,y,z\}\in L$. We prove that the algebra ${\cal L}_K(\Pi)$ is a Lie algebra if and only if $K$ is of even characteristic and $\Pi$ is a cotriangular space satisfying Pasch's axiom. Moreover, if $\Pi$ is a cotriangular space satisfying Pasch's axiom, then a connection between derivations of the Lie algebra ${\cal L}_K(\Pi)$ and geometric hyperplanes of $\Pi$ is used to determine the structure of the algebra of derivations of ${\cal L}_K(\Pi)$.