## Revista Matemática Iberoamericana

### 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.

#### Article information

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

Dates
First available in Project Euclid: 8 September 2003

https://projecteuclid.org/euclid.rmi/1063050155

Mathematical Reviews number (MathSciNet)
MR2023187

Zentralblatt MATH identifier
1055.17003

#### Citation

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. https://projecteuclid.org/euclid.rmi/1063050155

#### References

• 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.