Editing Semisimple Lie algebra (section)

==Jordan decomposition==
Each [[endomorphism]] ''x'' of a finite-dimensional vector space over a field of characteristic zero can be decomposed uniquely into a [[semisimple operator|semisimple]] (i.e., diagonalizable over the algebraic closure) and [[nilpotent endomorphism|nilpotent]] part
:<math>x=s+n\ </math>
such that ''s'' and ''n'' commute with each other.  Moreover, each of ''s'' and ''n'' is a polynomial in ''x''.  This is the [[Jordan–Chevalley decomposition|Jordan decomposition]] of ''x''.

The above applies to the [[adjoint representation of a Lie algebra|adjoint representation]] <math>\operatorname{ad}</math> of a semisimple Lie algebra <math>\mathfrak g</math>. An element ''x'' of <math>\mathfrak g</math> is said to be semisimple (resp. nilpotent) if <math>\operatorname{ad}(x)</math> is a semisimple (resp. nilpotent) operator.<ref>{{harvnb|Serre|2000|loc=Ch. II, § 5. Definition 3.}}</ref> If <math>x\in\mathfrak g</math>, then the '''abstract Jordan decomposition''' states that ''x'' can be written uniquely as:
:<math>x = s + n</math>
where <math>s</math> is semisimple, <math>n</math> is nilpotent and <math>[s, n] = 0</math>.<ref>{{harvnb|Serre|2000|loc=Ch. II, § 5. Theorem 6.}}</ref> Moreover, if <math>y \in \mathfrak g</math> commutes with ''x'', then it commutes with both <math>s, n</math> as well.

The abstract Jordan decomposition factors through any representation of <math>\mathfrak g</math> in the sense that given any representation ρ,
:<math>\rho(x) = \rho(s) + \rho(n)\,</math>
is the Jordan decomposition of ρ(''x'') in the endomorphism algebra of the representation space.<ref>{{harvnb|Serre|2000|loc=Ch. II, § 5. Theorem 7.}}</ref> (This is proved as a consequence of [[Weyl's complete reducibility theorem]]; see [[Weyl's theorem on complete reducibility#Application: preservation of Jordan decomposition]].)