On Lie isomorphisms of rings
Oksana Bezushchak, Iryna Kashuba, Efim Zelmanov
TL;DR
The paper investigates when Lie isomorphisms between the Lie rings $[A,A]$ of associative rings are standard, extending Herstein-style results beyond semiprime and torsion-free contexts. By introducing root-graded Lie rings via three (or four) pairwise orthogonal full idempotents, the authors develop a universal specialization and a universal annihilator extension framework, enabling the extension of Lie isomorphisms to ring homomorphisms or to a standard combination of a homomorphism and an anti-homomorphism. They prove that under the presence of three orthogonal full idempotents (with suitable fullness conditions), every Lie isomorphism $[A,A] o[B,B]$ is standard, and similarly extend to non-unital rings through annihilator extensions and universal central extensions. The results yield characteristic-free descriptions of automorphisms and derivations for Lie algebras of infinite matrices, and include a sharp non-standard example showing the necessity of the triple-idempotent hypothesis. Overall, the work provides a unified, algebraic framework that connects Lie isomorphisms with associative ring homomorphisms via universal constructions, with broad implications for structure theory of Lie algebras arising from rings.
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.