Revista Matemática Iberoamericana


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Volume 19, Issue 2, 2003, pp. 325–338
DOI: 10.4171/RMI/349

Published online: 2003-08-31

Graphs associated with nilpotent Lie algebras of maximal rank

Eduardo Diáz[1], Rafael Fernández-Mateos[2], Desemparados Fernández-Ternero[3] and Juan Núñez[4]

(1) Universidad de Sevilla, Spain
(2) Universidad de Sevilla, Spain
(3) Universidad de Sevilla, Spain
(4) Universidad de Sevilla, Spain

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.

Keywords: Nilpotent, maximal rank, Kac-Moody algebra, directed graph

Diáz Eduardo, Fernández-Mateos Rafael, Fernández-Ternero Desemparados, Núñez Juan: Graphs associated with nilpotent Lie algebras of maximal rank. Rev. Mat. Iberoam. 19 (2003), 325-338. doi: 10.4171/RMI/349