Semisimple Lie algebra

A finite-dimensional Lie algebra $\mathfrak g$ (over a field of characteristic $0$) is semisimple if it has no nonzero solvable ideals. Equivalently, if $\mathrm{Rad}(\mathfrak g)$ denotes the solvable radical, then $\mathfrak g$ is semisimple exactly when $\mathrm{Rad}(\mathfrak g)=0$ (compare solvable Lie algebra and ideals).

A fundamental characterization is:

• Cartan’s criterion / Killing form test: in characteristic $0$, $\mathfrak g$ is semisimple if and only if its Killing form is nondegenerate (see Killing form nondegenerate iff semisimple, and compare Cartan’s criterion).

Semisimple Lie algebras are the “non-abelian, non-solvable core” of general Lie algebras. The Levi decomposition expresses any finite-dimensional Lie algebra as a semidirect product of a semisimple Lie algebra with its solvable radical. Internally, every semisimple Lie algebra splits as a direct sum of simple ideals (see semisimple as a direct sum of simple ideals).

For complex semisimple Lie algebras, additional structure is encoded by roots (see roots of a Lie algebra) and the associated root system.