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

that consists of all linear combinations of Lie brackets of pairs of elements of The derived series is the sequence of subalgebras

If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is called solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory, and solvable Lie algebras are analogs of solvable groups. Any nilpotent Lie algebra is a fortiori solvable 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. The solvable Lie algebras are precisely those that can be obtained from semidirect products, starting from 0 and adding one dimension at a time. (v) For dimensional, there is a finite sequence of subalgebras of :

with each an ideal in A sequence of this type is called an elementary sequence. (vi) There is a finite sequence of subalgebras of ,

such that is an ideal in and is abelian. (vii) The Killing form of satisfies for all X in and Y in This is Cartan's criterion for solvability.

A Lie algebra is called completely solvable or split solvable if it has an elementary sequence{(V) As above definition} of ideals in from to A finite dimensional nilpotent Lie algebra is completely solvable, and a completely solvable Lie algebra is solvable. Over an algebraically closed field a solvable Lie algebra is completely solvable, but the dimensional real Lie algebra of the group of Euclidean isometries of the plane is solvable but not completely solvable. A solvable Lie algebra is split solvable if and only if the eigenvalues of are in for all in .

Every abelian Lie algebra is solvable by definition, since its commutator This includes the Lie algebra of diagonal matrices in , which are of the formfor The Lie algebra structure on a vector space given by the trivial bracket for any two matrices gives another example.

Another class of examples comes from nilpotent Lie algebras since the adjoint representation is solvable. Some examples include the upper diagonal matrices, such as the class of matrices of the formcalled the Lie algebra of strictly upper triangular matrices. In addition, the Lie algebra of upper diagonal matrices in form a solvable Lie algebra. This includes matrices of the formand is denoted .

Let be the set of matrices on the formThen is solvable, but not split solvable. It is isomorphic with the Lie algebra of the group of translations and rotations in the plane.

A semisimple Lie algebra is never solvable since its radical , which is the largest solvable ideal in , is trivial. page 11