Equivalent characterizations of semisimplicity for Lie algebras

Theorem (TFAE: semisimplicity)

Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$ (in particular over $\mathbb R$ or $\mathbb C$). The following are equivalent.

1. No nonzero solvable ideals: $\mathfrak g$ is semisimple , i.e. it has no nonzero solvable ideal (equivalently, its radical is $0$).

2. Nondegenerate Killing form: the Killing form $\kappa(X,Y)=\mathrm{tr}(\mathrm{ad}_X\mathrm{ad}_Y)$ is nondegenerate; compare nondegeneracy of the Killing form .

3. Direct sum of simple ideals: $\mathfrak g$ is a (finite) direct sum of simple Lie algebras ; see semisimple equals direct sum of simple ideals .

4. Adjoint representation is completely reducible: the representation $\mathrm{ad}:\mathfrak g\to \mathfrak{gl}(\mathfrak g)$ is completely reducible .

5. Cartan-type trace criterion: $\mathfrak g$ satisfies a trace criterion equivalent to semisimplicity as in Cartan’s criterion for semisimplicity .

Context

These equivalences are used interchangeably in practice: (2) is often the fastest computational test, while (3) is the structural starting point for classification (compare classification of simple Lie algebras ). Complete reducibility of representations (see Weyl’s theorem ) is a major consequence of semisimplicity.