Group gradings on finitary simple Lie algebras

TL;DR

This paper classifies all possible gradings by any abelian group on certain infinite-dimensional simple Lie algebras of linear transformations, expanding understanding of their structural symmetries.

Contribution

It provides a comprehensive classification of gradings on finitary simple Lie algebras, a previously uncharted area in infinite-dimensional Lie theory.

Findings

Complete classification of abelian group gradings on finitary simple Lie algebras.

Identification of isomorphism classes of these gradings.

Extension of known results from finite to infinite-dimensional cases.

Abstract

We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.