Abelian gradings in Lie algebras

TL;DR

This paper proves that a Lie algebra graded by a group without orthogonal graded ideals and generated by its support must have an abelian grading group, revealing structural constraints on such gradings.

Contribution

It establishes a new condition under which the grading group of a Lie algebra is necessarily abelian, linking algebraic properties to group-theoretic structure.

Findings

If a Lie algebra has no orthogonal graded ideals and is generated by its support, then its grading group is abelian.

The result connects the absence of orthogonal ideals with the abelian property of the grading group.

Provides a criterion for the abelian nature of grading groups in Lie algebra theory.

Abstract

Given a Lie algebra $L$ graded by a group $G$, if $L$ is does not contain orthogonal graded ideals and $G$ is generated by the support of $L$, then $G$ is an abelian group.