Graphs associated with nilpotent Lie algebras of maximal rank. (English) Zbl 1055.17003

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

Rev. Mat. Iberoam. 19, No. 2, 325-338 (2003).

A nilpotent Lie algebra \({\mathfrak L}\) has maximal rank if there exists an Abelian Lie algebra \(T\) of semisimple derivations of \({\mathfrak L}\) such that \(\dim T=\dim({\mathfrak L}/[{\mathfrak L},{\mathfrak L}])\). If this is the case, then one can construct a graph whose vertices are the roots of the natural action of \(T\) on \({\mathfrak L}\) and whose edges describe the commutation relations between the root spaces. The aim of the paper under review is just to describe in detail the corresponding construction.
The paper also includes a number of theorems to the effect that the classification of nilpotent Lie algebras of maximal rank is thus reduced to a classification problem on the subgraphs of a certain graph.

Reviewer: Daniel Beltiţă (Bucureşti)

MSC:

17B30	Solvable, nilpotent (super)algebras
05C20	Directed graphs (digraphs), tournaments
05C85	Graph algorithms (graph-theoretic aspects)

Keywords:

nilpotent Lie algebra; maximal rank; directed graph