Bulletin of the Belgian Mathematical Society - Simon Stevin

Lie Algebras and Cotriangular Spaces

Hans Cuypers

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

Article information

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

Dates
First available in Project Euclid: 3 June 2005

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1117805084

Mathematical Reviews number (MathSciNet)
MR2179964

Zentralblatt MATH identifier
1119.05018

Citation

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


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