On Lie isomorphisms of rings

TL;DR

This paper investigates conditions under which Lie ring isomorphisms of associative rings are standard, extending to non-unital rings and applying results to automorphisms and derivations of infinite matrix Lie algebras.

Contribution

It establishes that under certain idempotent decompositions, Lie ring isomorphisms are standard, and extends these results to non-unital rings with specific assumptions.

Findings

Lie isomorphisms are standard when the identity decomposes into orthogonal idempotents.

Extension of Lie isomorphisms to ring homomorphisms under certain conditions.

Application to automorphisms and derivations of infinite matrix Lie algebras.

Abstract

An associative ring $A$ gives rise to the Lie ring $A^{(-)}=(A,[a,b ]=ab-ba)$. The subject of isomorphisms of Lie rings $A^{(-)}$ and $[A,A]$ has attracted considerable attention in the literature. We prove that if the identity element of $A$ decomposes into a sum of at least three full orthogonal idempotents, then any isomorphism from the Lie ring $[A,A]$ to the Lie ring $[B,B]$ is standard. For non-unital rings, the description is more intricate. Under a certain assumption on idempotents, we extend a Lie isomorphism from $[A,A]$ to $[B,B]$ to a homomorphism of associative rings $\widehat{A\oplus A^{op}}\to B,$ where $A^{op}=(A,a\cdot b= b\cdot a),$ and $\widehat{A\oplus A^{op}}\to A\oplus A^{op}$ is the universal annihilator extension of the ring $A\oplus A^{op}.$ The results obtained are then applied to the description of automorphisms and derivations of Lie algebras of infinite matrices.