Defects in weighted graphs and commutators

Abstract

Let R be a commutative ring. In [5], 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 a consistent labeling 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 the 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 derived subalgebra $L^{\prime }$ is at most 3-dimensional, the Lie bracket is surjective onto $L^{\prime }$. We provide a counterexample when $\dim \left(L^{\prime }\right)=4$. We also characterize commutators among the elements of L' when $\dim (L/Z(L\left)\right)\le 4$.