Revista Matemática Iberoamericana

Graphs associated with nilpotent Lie algebras of maximal rank

Eduardo Díaz, Rafael Fernández-Mateos, Desamparados Fernández-Ternero, and Juan Núñez

Full-text: Open access


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 information

Rev. Mat. Iberoamericana, Volume 19, Number 2 (2003), 325-338.

First available in Project Euclid: 8 September 2003

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 32S20: Global theory of singularities; cohomological properties [See also 14E15] 32S45: Modifications; resolution of singularities [See also 14E15] 05C20: Directed graphs (digraphs), tournaments 05C85: Graph algorithms [See also 68R10, 68W05] 17B30: Solvable, nilpotent (super)algebras 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65]

nilpotent maximal rank Kac-Moody algebra directed graph


Díaz, Eduardo; Fernández-Mateos, Rafael; Fernández-Ternero, Desamparados; Núñez, Juan. Graphs associated with nilpotent Lie algebras of maximal rank. Rev. Mat. Iberoamericana 19 (2003), no. 2, 325--338.

Export citation


  • Favre, G.: Système de poids sur une algèbre de Lie nilpotente. Manuscripta Math. 9 (1973), 53-90.
  • Fernández-Ternero, D.: Clasificación de álgebras de Lie nilpotentes de rango maximal. Tesis Doctoral, Universidad de Sevilla, 2001.
  • Kac, V. G.: Infinite dimensional Lie algebras (3rd ed). Cambridge University Press, Cambridge, 1990.
  • König, D.: Theory of Finite and Infinite Graphs. Birkhäuser Boston, 1990.
  • Santharoubane, L. J.: Kac-Moody Lie algebras and the classification of nilpotent Lie algebras of maximal rank. Canad. J. Math. 34 (1982), 1215-1239.