In mathematics, a Lie algebra
is solvable if its derived series terminates in the zero subalgebra.
, 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.
[1] 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.
The largest solvable ideal of a Lie algebra is called the radical.
be a finite-dimensional Lie algebra over a field of characteristic 0.
Lie's Theorem states that if
is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and
is a solvable Lie algebra, and if
is called completely solvable or split solvable if it has an elementary sequence{(V) As above definition} of ideals in
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
Every abelian Lie algebra
is solvable by definition, since its commutator
This includes the Lie algebra of diagonal matrices in
The Lie algebra structure on a vector space
Another class of examples comes from nilpotent Lie algebras since the adjoint representation is solvable.
called the Lie algebra of strictly upper triangular matrices.
In addition, the Lie algebra of upper diagonal matrices in
form a solvable Lie algebra.
[2] It is isomorphic with the Lie algebra of the group of translations and rotations in the plane.
A semisimple Lie algebra
, which is the largest solvable ideal in
[1] page 11 Because the term "solvable" is also used for solvable groups in group theory, there are several possible definitions of solvable Lie group.