Jordan-Chevalley decomposition in finite dimesional Lie algebras

TL;DR

This paper characterizes when elements of finite-dimensional Lie algebras have an abstract Jordan-Chevalley decomposition, linking it to the algebra being perfect and providing explicit descriptions of the decomposition.

Contribution

It proves that an element has an abstract Jordan-Chevalley decomposition if and only if it lies in the derived algebra, and it shows this is equivalent to the algebra being perfect, using elementary methods.

Findings

Elements in [g,g] have unique Jordan-Chevalley decompositions.

A Lie algebra is perfect if and only if all elements have such decompositions.

In subalgebras of gl(n,k), the semisimple and nilpotent parts lie in the derived algebra.

Abstract

Let $\g$ be a finite dimensional Lie algebra over a field $k$ of characteristic zero. An element $x$ of $\g$ is said to have an \emph{abstract Jordan-Chevalley decomposition} if there exist unique $s,n\in\g$ such that $x=s+n$, $[s,n]=0$ and given any finite dimensional representation $\pi:\g\to\gl(V)$ the Jordan-Chevalley decomposition of $\pi(x)$ in $\gl(V)$ is $\pi(x)=\pi(s)+\pi(n)$. In this paper we prove that $x\in\g$ has an abstract Jordan-Chevalley decomposition if and only if $x\in [\g,\g]$, in which case its semisimple and nilpotent parts are also in $[\g,\g]$ and are explicitly determined. We derive two immediate consequences: (1) every element of $\g$ has an abstract Jordan-Chevalley decomposition if and only if $\g$ is perfect; (2) if $\g$ is a Lie subalgebra of $\gl(n,k)$ then $[\g,\g]$ contains the semisimple and nilpotent parts of all its elements. The last result was first proved by Bourbaki using different methods. Our proof only uses elementary linear algebra and basic results on the representation theory of Lie algebras, such as the Invariance Lemma and Lie's Theorem, in addition to the fundamental theorems of Ado and Levi.