Defects in weighted graphs and Commutators

TL;DR

The paper introduces $F$-weighted graphs to encode balance equations $x_i y_j - x_j y_i = d_{i,j}$ and studies when these graphs admit consistent labelings, revealing a precise defect-based obstruction theory for four-vertex graphs. It proves that defectless four-vertex graphs are exactly the ones admitting consistent labelings and uses this to derive surjectivity results for Lie brackets on countable-dimensional Lie algebras, showing $L'=[L,L]$ whenever $\dim(L')\le 3$ and providing a counterexample when $\dim(L')=4$. The authors then extend the framework to $p$-groups of class $2$ and exponent $p$, establishing corresponding criteria for when a given commutator lies in the commutator subgroup $K(G)$. Throughout, they connect graph-theoretic defects to algebraic surjectivity and commutativity properties, yielding a unified approach to commutators in both Lie algebras and nilpotent groups and a complete characterization in the case $\dim(L/Z(L))\le 4$.

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 $\dim(L/Z(L))\leq 4$.

Figures (6)

• Figure 1: Defect $(4)_A$: Graph $\Gamma_1$ with $d_{1,2} \ne 0$, $d_{3,4} \ne 0$ and $d_{1,3}=d_{2,3}=0$.
• Figure 2: Defect $(4)_B$: Graph $\Gamma_2$ with $d_{1,2}d_{1,4}d_{2,3}d_{3,4} \ne 0$, $d_{1,3}=0$ and $d_{1,2}d_{3,4} \ne -d_{1,4}d_{2,3}$.
• Figure 3: Defect $(4)_C$: Graph $\Gamma_3$ with $d_{1,2}d_{1,3}d_{1,4}d_{2,3}d_{2,4}d_{3,4} \ne 0$ and $d_{1,2}d_{3,4} + d_{1,4}d_{2,3} \ne d_{1,3}d_{2,4}$.
• Figure 4: Connected graphs in $4$ vertices
• Figure 5: Graph $\Gamma$ with $d_{1,5}d_{2,5}d_{3,4}\ne 0$ and $d_{1,2} = d_{1,4} = d_{2,3} = 0$.

Theorems & Definitions (27)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Lemma 2.1: Theorem 3.4,KK_2025(1)
• Lemma 2.2
• proof
• Lemma 2.3
• proof