Jordan-Chevalley decomposition in finite dimesional Lie algebras

LEANDRO CAGLIERO; FERNANDO SZECHTMAN

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Año: 2011 vol. 139 p. 3909 - 3913

0002-9939

Let $g$ be a finite dimensional Lie algebra overa field $k$ of characteristic zero.An element $x$ of $g$ is said to have an emph{abstract Jordan-Chevalley decomposition}if there exist unique $s,ning$ such that $x=s+n$, $[s,n]=0$ and given anyfinite dimensional representation $pi:g ogl(V)$the Jordan-Chevalley decomposition of $pi(x)$ in $gl(V)$ is $pi(x)=pi(s)+pi(n)$.In this paper we prove that $xing$ has an abstract Jordan-Chevalley decompositionif and only if $xin [g,g]$, in which case its semisimple and nilpotent parts arealso 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 ofall its elements. The last result was first proved by Bourbaki using different methods.Our proof only useselementary linear algebra and basic results on the representation theoryof Lie algebras, such as the Invariance Lemma and Lie´s Theorem,in addition to the fundamental theorems of Ado and Levi.