Solvable Lie algebra

In mathematics, a Lie algebra g is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra is the subalgebra of g, denoted

that consists of all Lie brackets of pairs of elements of g. The derived series is the sequence of subalgebras

If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory.

Any nilpotent Lie algebra is solvable, a fortiori, but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition.

A maximal solvable subalgebra is called a Borel subalgebra. The largest solvable ideal of a Lie algebra is called the radical.

Characterizations

Let g be a finite-dimensional Lie algebra over a field of characteristic 0. The following are equivalent.