A note on the structure of graded Lie algebras

TL;DR

This paper provides explicit upper bounds on the derived length and rank of certain Lie algebra components, extending classical theorems by analyzing automorphisms and fixed points.

Contribution

It offers a short, simple proof for bounds on the radical's derived length and Levi complement's rank in complex Lie algebras with automorphisms.

Findings

Upper bounds for derived length of the radical R

Bounds for the rank of the Levi complement G/R

Extension of classical theorems by Kreknin, Shalev, and Jacobson

Abstract

Consider a finite-dimensional, complex Lie algebra G and a semi-simple automorphism {\alpha}. This note aims to give a short and simple proof for explicit upper bounds for the derived length of the radical R and the rank of a Levi complement G/R in terms of the number of eigenvalues of {\alpha} and the dimension of the space of fixed-points. This is an extension of classical theorems by Kreknin, Shalev and Jacobson.