radical of a Lie algebra

Definition

For a finite dimensional Lie algebra , the unique solveable ideal containing all solvable ideals in is called the radical of . It is often denoted .

Existence

If is a finite dimension Lie algebra, then exists.

Proof

Let be a solvable ideal of with maximal dimension.

Claim: if and are solvable ideals of , then is solvable.

Subproof: Consider the Lie (sub)algebra . Then is a solvable ideal of . By the second isomorphism theorem, we have

So since and are solvable, then is solvable. Solvability is preserved under extension, so is solvable as well.

Therefore, consider a solvable ideal such that . Then is solvable with dimension larger than which contradicts the maximality of . Therefore, for all solvable ideals .

Relation to semisimple Lie algebras

For a finite dimensional Lie algebra , the quotient Lie algebra is semisimple.

Proof

Let be a solvable ideal, it suffices to show . Consider the quotient homomorphism

is an idea since

We next show that is solvable in . By extension we have

By assumption, is solvable and which is solvable (since is solvable). Therefore, is solvable and so .