A note on the Jordan decomposition

TL;DR

This paper proves that Jordan components of endomorphisms and isomorphisms in finite-dimensional vector spaces are polynomial functions, and shows these components lie within the algebra or group for semisimple Lie structures, linking linear and abstract decompositions.

Contribution

It establishes that Jordan components are polynomial in the original element and confirms their presence in the algebra or group for semisimple Lie structures, unifying linear and abstract decompositions.

Findings

Jordan components are polynomial functions of the original element.

Jordan components lie within the algebra or group for semisimple Lie structures.

Linear and abstract Jordan decompositions coincide in this context.

Abstract

In this article we prove that the elliptic, hyperbolic and nilpotent (or unipotent) additive (or multiplicative) Jordan components of an endomorphism $X$ (or an isomorphism $g$) of a finite dimensional vector space are given by polynomials in $X$ (or in $g$). By using this, we provide a simple proof that, for an element $X$ of a linear semisimple Lie algebra $\g$ (or $g$ of a linear semisimple connected Lie group $G$), its three Jordan components lie again in the algebra (in the group). This was previously unknown for linear Lie groups other then $\Int(\g)$. This implies that, for this class of algebras and groups, the usual linear Jordan decomposition coincides with the abstract Jordan decomposition.