Graphs associated with nilpotent Lie algebras of maximal rank

  • Eduardo Diáz

    Universidad de Sevilla, Spain
  • Rafael Fernández-Mateos

    Universidad de Sevilla, Spain
  • Desemparados Fernández-Ternero

    Universidad de Sevilla, Spain
  • Juan Núñez

    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 AA and it is isomorphic to a quotient of the positive part n+\mathfrak{n}_+ of the Kac-Moody algebra g(A)\mathfrak{g}(A). Then, if AA is affine, we can associate n+\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 AA. Finally, we show an algorithm which obtains these subgraphs and also groups them in isomorphism classes.

Cite this article

Eduardo Diáz, Rafael Fernández-Mateos, Desemparados Fernández-Ternero, Juan Núñez, Graphs associated with nilpotent Lie algebras of maximal rank. Rev. Mat. Iberoam. 19 (2003), no. 2, pp. 325–338

DOI 10.4171/RMI/349