Jordan-Chevalley decomposition in finite dimensional Lie algebras

Abstract:

Let $\mathfrak {g}$ be a finite dimensional Lie algebra over a field $k$ of characteristic zero. An element $x$ of $\mathfrak {g}$ is said to have an abstract Jordan-Chevalley decomposition if there exist unique $s,n\in \mathfrak {g}$ such that $x=s+n$, $[s,n]=0$ and given any finite dimensional representation $\pi :\mathfrak {g}\to \mathfrak {gl}(V)$ the Jordan-Chevalley decomposition of $\pi (x)$ in $\mathfrak {gl}(V)$ is $\pi (x)=\pi (s)+\pi (n)$.

In this paper we prove that $x\in \mathfrak {g}$ has an abstract Jordan-Chevalley decomposition if and only if $x\in [\mathfrak {g},\mathfrak {g}]$, in which case its semisimple and nilpotent parts are also in $[\mathfrak {g},\mathfrak {g}]$ and are explicitly determined. We derive two immediate consequences: (1) every element of $\mathfrak {g}$ has an abstract Jordan-Chevalley decomposition if and only if $\mathfrak {g}$ is perfect; (2) if $\mathfrak {g}$ is a Lie subalgebra of $\mathfrak {gl}(n,k)$, then $[\mathfrak {g},\mathfrak {g}]$ contains the semisimple and nilpotent parts of all its elements. The last result was first proved by Bourbaki using different methods.

Our proof uses only 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.