On Dynkin gradings in simple Lie algebras

TL;DR

This paper explores gradings in simple Lie algebras induced by nilpotent elements, focusing on abelian subalgebras of degree 1 and establishing a canonical reduction to strictly odd nilpotent elements.

Contribution

It introduces a canonical reduction technique for studying gradings from nilpotent elements, emphasizing the role of strictly odd nilpotent elements in simple Lie algebras.

Findings

Existence of canonical associated strictly odd nilpotent elements for odd nilpotent elements

Reduction of grading investigations to strictly odd nilpotent elements

Enhanced understanding of abelian subalgebras in Lie algebra gradings

Abstract

In this paper we study gradings on simple Lie algebras arising from nilpotent elements. Specifically, we investigate abelian subalgebras which are degree 1 homogeneous with respect to these gradings. We show that for each odd nilpotent element there always exists canonically associated "strictly" odd nilpotent element, which allows us to reduce our investigations to the latter.