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June 2005 Lie Algebras and Cotriangular Spaces
Hans Cuypers
Bull. Belg. Math. Soc. Simon Stevin 12(2): 209-221 (June 2005). DOI: 10.36045/bbms/1117805084


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


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Hans Cuypers. "Lie Algebras and Cotriangular Spaces." Bull. Belg. Math. Soc. Simon Stevin 12 (2) 209 - 221, June 2005.


Published: June 2005
First available in Project Euclid: 3 June 2005

zbMATH: 1119.05018
MathSciNet: MR2179964
Digital Object Identifier: 10.36045/bbms/1117805084

Rights: Copyright © 2005 The Belgian Mathematical Society


Vol.12 • No. 2 • June 2005
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