On the Lie structure of locally matrix algebras

TL;DR

This paper characterizes when the Lie algebra derived from a unital locally matrix algebra is simple and when its derivation algebra is topologically simple, based solely on the Steinitz number of the algebra.

Contribution

It provides necessary and sufficient conditions for the simplicity of the Lie algebra and its derivations in terms of the Steinitz number, advancing understanding of their structure.

Findings

Lie algebra $A/\mathbb{F}\cdot 1$ is simple under certain conditions

Derivation algebra $\text{Der}(A)$ is topologically simple under certain conditions

Conditions depend only on the Steinitz number of $A$

Abstract

Let $A$ be a unital locally matrix algebra over a field $\mathbb{F}$ of characteristic different from $2.$ We find necessary and sufficient condition for the Lie algebra $A\diagup\mathbb{F}\cdot 1$ to be simple and for the Lie algebra of derivations $\text{Der}(A)$ to be topologically simple. The condition depends on the Steinitz number of $A$ only.