solvable Lie algebra

Definition

A Lie algebra is called solvable if the derived series terminates. That is, if there exists such that .

Properties

Preserved under sub and quotient object

Let be a solvable Lie algebra, with a subalgebra and an ideal . The Lie algebras and are solvable.

Proof

Note for a subalgebra then . Therefore, the derived series will terminate for both Lie algebras.

Let be a surjective Lie algebra homomorphism. We have . Then is surjective by induction since

Therefore, if the series terminates for then by the homomorphism it will for the quotient as well.

Preserved under extension

Consider the following short exact sequence of Lie algebras:

Then if and are solvable, then is also solvable. This is equivalent to saying that solvability is preserved under Lie algebra extension.

Proof

Suppose is a solvable ideal and is solvable with quotient homomorphism . Consider the series . We have

since is a homomorphism.

Since is solvable, we have such that . Then , so . is solvable, so The ideal is solvable, so eventually its derived series will terminate. By induction, we have

Therefore, eventually the series will terminate and is solvable.

Examples

In , the subalgebra of upper-triangular matrices is solvable.

Any nilpotent Lie algebra is solvable.