Solvable Lie algebras and graphs

TL;DR

This paper introduces a class of solvable Lie algebras constructed from graphs, establishes their isomorphism criteria based on graph isomorphism, and explores their metric properties including examples of spaces with nonpositive curvature.

Contribution

It defines a new class of solvable Lie algebras from graphs and proves their isomorphism correspondence with graph isomorphism, also analyzing their metric properties.

Findings

Lie algebra isomorphism iff graph isomorphism

Examples of homogeneous spaces with nonpositive curvature

Introduction of solvable extensions based on 3-cliques

Abstract

We define a solvable extension of the graph 2-step nilpotent Lie algebras of [5] by adding elements corresponding to the 3-cliques of the graph. We study some of their basic properties and we prove that two such Lie algebras are isomorphic if and only if their graphs are isomorphic. We also briefly discuss some metric properties, providing examples of homogeneous spaces with nonpositive curvature operator and solvsolitons.