Uniqueness of addition in Lie algebras revisited

Abstract

We obtain new and improve old results on uniqueness of addition in Lie rings and Lie algebras. A Lie ring $\mathfrak{R}$ is called a unique addition ring, or a UA-Lie ring, if any commutator-preserving bijection from $\mathfrak{R}$ to an arbitrary Lie ring is additive. We describe wide classes of Lie rings that are not UA-Lie ring. In the other direction, it is known that if a finite-dimensional Lie algebra $\mathfrak{g}$ contains two elements whose centralizers have trivial intersection, then $\mathfrak{g}$ is a UA-Lie ring. We use this result to characterize UA-Lie rings among seaweed Lie algebras. The paper includes many open problems and questions.