Solvable Lie algebra

A Lie algebra whose derived series eventually becomes zero; the Lie-algebra analogue of solvable groups.

Solvable Lie algebra

Let $\mathfrak g$ be a Lie algebra. Define its derived series (see derived series ) by

\mathfrak g^{(0)}=\mathfrak g,\qquad \mathfrak g^{(k+1)}=[\mathfrak g^{(k)},\mathfrak g^{(k)}],

where $[\cdot,\cdot]$ denotes the Lie bracket and each derived algebra is an ideal (compare derived subalgebra and it is an ideal ).

The Lie algebra $\mathfrak g$ is solvable if there exists $n$ such that

\mathfrak g^{(n)}=0.

Equivalently, iterated commutators eventually vanish. This notion is central in the structure theory of general Lie algebras: the maximal solvable ideal is the radical, and the Levi decomposition splits any finite-dimensional Lie algebra (in characteristic $0$ ) into a semisimple part and a solvable part.

There are several important tests and relations:

Cartan’s criterion gives a practical characterization in many settings (see Cartan’s criterion for solvability and equivalent conditions for solvability ).
Every nilpotent Lie algebra is solvable (see nilpotent implies solvable ), but not conversely.