Uniform Lie Algebras and Uniformly Colored Graphs

TL;DR

This paper introduces uniform Lie algebras, explores their properties, classifies small cases, and establishes a correspondence with uniformly colored graphs, providing methods to construct infinite families of both structures.

Contribution

It defines uniformly colored graphs and establishes a correspondence with uniform Lie algebras, including classification and construction methods.

Findings

Classified uniform Lie algebras with up to five generators.

Established a correspondence between uniform Lie algebras and uniformly colored graphs.

Presented methods to construct infinite families of these structures.

Abstract

Uniform Lie algebras are combinatorially defined two-step nilpotent Lie algebras which can be used to define Einstein solvmanifolds. These Einstein spaces often have nontrivial isotropy groups. We derive basic properties of uniform Lie algebras and we classify uniform Lie algebras with five or fewer generators. We define a type of directed colored graph called a uniformly colored graph and establish a correspondence between uniform Lie algebras and uniformly colored graphs. We present several methods of constructing infinite families of uniformly colored graphs and corresponding uniform Lie algebras.