Defects in weighted graphs and Commutators

TL;DR

This paper explores the structure of weighted graphs over fields to understand commutators in Lie algebras and groups, introducing defects as a key concept that influences the surjectivity of Lie brackets and commutator elements.

Contribution

It introduces the notion of defects in weighted graphs over fields and establishes their impact on the surjectivity of Lie brackets and commutators, providing new criteria and counterexamples.

Findings

Defects prevent Lie brackets from being surjective onto the derived subalgebra.

For Lie algebras with dimension at most countable and derived subalgebra of dimension ≤ 3, the bracket is surjective.

Counterexample provided for derived subalgebra of dimension 4.

Abstract

Let $R$ be a commutative ring. In \cite{KK_2025(1)}, the authors introduced $R$-weighted graphs as a tool for studying commutators in groups and Lie algebras. These graphs are equivalent to a system of balance equations, and their consistent labelings correspond to solutions of this system of balance equations. In this article, we apply these ideas in the case when $R$ is a field $F$. We focus on $F$-weighted graphs with four vertices and establish necessary and sufficient conditions for the existence of consistent labelings on them. A notion of defects in weighted graphs is introduced for this purpose. We prove that defects in weighted graphs prevent Lie brackets from being surjective onto its derived Lie subalgebra. Similarly, these defects prevent certain elements in the commutator subgroup of a nilpotent group of class $2$ from being a commutator. As an application of our techniques, we prove that for a Lie algebra $L$ whose dimension over $F$ is at most countable and the dimension of its derived subalgebra $L'$ is at most $3$, the Lie bracket is surjective onto $L'$. We provide a counterexample when $\dim(L') = 4$. We also characterize commutators among $L$' for the Lie algebras $L$ with $\dim(L/Z(L))\leq 4$.