Abelian Factor in 2-step Nilpotent Lie Algebras Constructed from Graphs

TL;DR

This paper investigates how the structure of edge labels in graphs influences the abelian factor in associated 2-step nilpotent Lie algebras, providing explicit computations for various graph families and analyzing their singularity properties.

Contribution

It characterizes the abelian factor in graph-constructed 2-step nilpotent Lie algebras and computes it explicitly for specific graph families, extending previous work.

Findings

The structure of edge labeling affects the abelian factor in the Lie algebra.

Explicit formulas for the abelian factor in star, cycle, Schreier, and edge-colored graphs.

Analysis of singularity properties in certain cases.

Abstract

We consider real 2-step metric nilpotent Lie algebras associated to graphs with possibly repeated edge labels as constructed by Ray in 2016. We determine how the structure of the egde labeling within the graph contributes to the abelian factor in these Lie algebras. Furthermore, we explicitly compute the abelian factor of the 2-step nilpotent Lie algebras associated with some families of graphs such as star graphs, cycles, Schreier graphs, and properly edge-colored graphs. We also study the singularity properties of these Lie algebras in certain cases.