Jordan-Chevalley decomposition in Lie algebras

TL;DR

This paper investigates conditions under which the Jordan-Chevalley decomposition components of a matrix in a solvable Lie algebra also belong to that algebra, providing a characterization involving the existence of specific semisimple and nilpotent elements.

Contribution

It establishes a necessary and sufficient condition for the Jordan-Chevalley components to lie within a solvable Lie algebra based on the existence of particular semisimple and nilpotent elements.

Findings

The semisimple and nilpotent parts of the Jordan-Chevalley decomposition are in the algebra if and only if certain elements exist.

Provides a characterization of when the Jordan-Chevalley decomposition components belong to the algebra.

The result applies to solvable Lie algebras of matrices over fields of characteristic zero.

Abstract

We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0, and $A\in\mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan-Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$) such that $A=S+N$.