Revista Matemática Iberoamericana

Graphs associated with nilpotent Lie algebras of maximal rank

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

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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

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1063050155

Mathematical Reviews number (MathSciNet)
MR2023187

Zentralblatt MATH identifier
1055.17003

Subjects
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]

Keywords
nilpotent maximal rank Kac-Moody algebra directed graph

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


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References

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