Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals.)

Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra ${\displaystyle {\mathfrak {g}}}$, if nonzero, the following conditions are equivalent:

The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple.

Semisimple Lie algebras have a very elegant classification, in stark contrast to solvable Lie algebras. Semisimple Lie algebras over an algebraically closed field of characteristic zero are completely classified by their root system, which are in turn classified by Dynkin diagrams. Semisimple algebras over non-algebraically closed fields can be understood in terms of those over the algebraic closure, though the classification is somewhat more intricate; see real form for the case of real semisimple Lie algebras, which were classified by Élie Cartan.

Further, the representation theory of semisimple Lie algebras is much cleaner than that for general Lie algebras. For example, the Jordan decomposition in a semisimple Lie algebra coincides with the Jordan decomposition in its representation; this is not the case for Lie algebras in general.

If ${\displaystyle {\mathfrak {g}}}$ is semisimple, then ${\displaystyle {\mathfrak {g}}=[{\mathfrak {g}},{\mathfrak {g}}]}$. In particular, every linear semisimple Lie algebra is a subalgebra of ${\displaystyle {\mathfrak {sl}}}$, the special linear Lie algebra. The study of the structure of ${\displaystyle {\mathfrak {sl}}}$ constitutes an important part of the representation theory for semisimple Lie algebras.

The semisimple Lie algebras over the complex numbers were first classified by Wilhelm Killing (1888–90), though his proof lacked rigor. His proof was made rigorous by Élie Cartan (1894) in his Ph.D. thesis, who also classified semisimple real Lie algebras. This was subsequently refined, and the present classification by Dynkin diagrams was given by then 22-year-old Eugene Dynkin in 1947. Some minor modifications have been made (notably by J. P. Serre), but the proof is unchanged in its essentials and can be found in any standard reference, such as (Humphreys 1972).

Each endomorphism x of a finite-dimensional vector space over a field of characteristic zero can be decomposed uniquely into a semisimple (i.e., diagonalizable over the algebraic closure) and nilpotent part