Uniqueness of addition in Lie rings $\mathfrak{gl}_n(K)$ and $\mathfrak{sl}_n(K)$

Abstract

We prove that for any Lie ring $\mathfrak{R}$ and any commutator-preserving bijection $\alpha : \mathfrak{gl}_n(K) \rightarrow \mathfrak{R}$, the map $\alpha$ is additive on $\mathfrak{sl}_n(K)$, where $n \ge 2$ and $K$ is an arbitrary field. Using this result we find criteria for Lie rings $\mathfrak{gl}_n(K)$ and $\mathfrak{sl}_n(K)$ to be unique addition. We also show that any commutator-preserving injection of Lie rings $\beta : \mathfrak{sl}_2(K) \rightarrow \mathfrak{S}$ is additive. This is the first result on additivity of commutator-preserving injections.