Bulletin of the Belgian Mathematical Society - Simon Stevin

Lie Algebras and Cotriangular Spaces

Hans Cuypers

Full-text: Open access


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 ${\cal L}_K(\Pi)$.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 2 (2005), 209-221.

First available in Project Euclid: 3 June 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Cuypers, Hans. Lie Algebras and Cotriangular Spaces. Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 2, 209--221. https://projecteuclid.org/euclid.bbms/1117805084

Export citation