A note on random samples of Lie algebras

TL;DR

This paper discusses the properties of Lie algebras generated by solving linear systems for structure constants, highlighting their solvability and abelian derived subalgebras, and introduces an efficient numerical algorithm for their computation.

Contribution

It extends previous work by analyzing the algebraic properties of generated Lie algebras and providing a practical numerical method for calculating their structure constants.

Findings

Generated Lie algebras are solvable.

Derived subalgebras are Abelian.

An efficient numerical algorithm is proposed.

Abstract

Recently, Paiva and Teixeira (arXiv:1108.4396) showed that the structure constants of a Lie algebra are the solution of a system of linear equations provided a certain subset of the structure constants are given a-priori. Here it is noted that Lie algebras generated in this way are solvable and their derived subalgebras are Abelian if the system of linear equations considered by Paiva and Teixeira is not degenerate. An efficient numerical algorithm for the calculation of their structure constants is described.