$R$-weighted graphs and commutators

TL;DR

This paper introduces balance equations over commutative rings and associates them with weighted graphs, providing explicit solutions in certain cases and applying these to group theory and Lie algebra problems.

Contribution

It develops a novel framework linking balance equations over rings to graph labelings and applies this to analyze commutator structures in groups and Lie algebras.

Findings

Explicit solutions for balance equations over local rings.

Conditions for p-groups of class 2 to have their commutators equal to the commutator subgroup.

Results on the surjectivity of the Lie bracket in Lie algebras.

Abstract

In this article, we introduce balance equations over commutative rings $R$ and associate $R$-weighted graphs to them so that solving balance equations corresponds to a consistent labeling of vertices of the associated graph. Our primary focus is the case when $R$ is a commutative local ring whose residue field contains at least three elements. In this case, we provide explicit solutions of balance equations. As an application, letting $R$ to be the ring of $p$-adic integers, we examine some necessary and sufficient conditions for a $p$-group of nilpotency class $2$ to have its set of commutators coincide with its commutator subgroup. We also apply our results to study the surjectivity of the Lie bracket in Lie algebras, without any restriction on their dimension and the underlined field.