Graphs associated with nilpotent Lie algebras of maximal rank

Abstract

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.