Lie Algebras and Cotriangular Spaces

Cuypers, Hans

Cuypers, Hans

Bull. Belg. Math. Soc. Simon Stevin, Tome 11 (2005) no. 5, p. 209-221 / Harvested from Project Euclid

Text on Project Euclid
PDF
Alt PDF

Résumé

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

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

Publié le : 2005-04-14
Classification:

@article{1117805084,
     author = {Cuypers, Hans},
     title = {Lie Algebras and Cotriangular Spaces},
     journal = {Bull. Belg. Math. Soc. Simon Stevin},
     volume = {11},
     number = {5},
     year = {2005},
     pages = { 209-221},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1117805084}
}

Cuypers, Hans. Lie Algebras and Cotriangular Spaces. Bull. Belg. Math. Soc. Simon Stevin, Tome 11 (2005) no. 5, pp.  209-221. http://gdmltest.u-ga.fr/item/1117805084/