jordan decomposition

Last updated January 27, 2022

Idea

Qualitatively, the Jordan decomposition says that any endomorphism $X$ of a complex vector space may be expressed as the sum of a diagonalizable and a nilpotent endomorphism, and moreover these commute and may be expressed as polynomials in $X$: $$ X=X_s+X_n$$

Representations of lie algebras

Representation of a group lie algebra Semisimplicity

Let $\mathfrak{g}$ be a semisimple Lie algebra. For any $X\in\mathfrak{g}$, there exist $X_s$ and $X_n\in\mathfrak{g}$ such that for any representation $\rho:\mathfrak{g}\to\mathfrak{gl}(V)$ we have $$\rho(X)_s=\rho(X_s)\quad\text{and}\quad\rho(X)_n=\rho(X_n).$$