In this paper, we use the graphs as a tool to study nilpotent Lie
algebras. It implies to set up a link between graph theory and Lie
theory. To do this, it is already known that every nilpotent Lie
algebra of maximal rank is associated with a generalized Cartan
matrix $A$ and it is isomorphic to a quotient of the positive part
$\mathfrak{n}_+$ of the Kac-Moody algebra $\mathfrak{g}(A)$. Then,
if $A$ is affine, we can associate $\mathfrak{n}_+$ with a
directed graph (from now on, we use the term digraph) and we can
also associate a subgraph of this digraph with every isomorphism
class of nilpotent Lie algebras of maximal rank and of type $A$.
Finally, we show an algorithm which obtains these subgraphs and
also groups them in isomorphism classes.
@article{1063050155,
author = {D\'\i az, Eduardo and Fern\'andez-Mateos, Rafael and Fern\'andez-Ternero, Desamparados and N\'u\~nez, Juan},
title = {Graphs associated with nilpotent Lie algebras of maximal rank},
journal = {Rev. Mat. Iberoamericana},
volume = {19},
number = {2},
year = {2003},
pages = { 325-338},
language = {en},
url = {http://dml.mathdoc.fr/item/1063050155}
}
Díaz, Eduardo; Fernández-Mateos, Rafael; Fernández-Ternero, Desamparados; Núñez, Juan. Graphs associated with nilpotent Lie algebras of maximal rank. Rev. Mat. Iberoamericana, Tome 19 (2003) no. 2, pp. 325-338. http://gdmltest.u-ga.fr/item/1063050155/