Solvable Lie algebra

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

that consists of all 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 solvable.[1] The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory.

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.

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

Characterizations

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

with each an ideal in .[3] A sequence of this type is called an elementary sequence.
such that is an ideal in and is abelian.[4]

Properties

Lie's Theorem states that if is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and is a solvable linear Lie algebra over a subfield of , and if is a representation of over , then there exists a simultaneous eigenvector of the matrices for all elements . More generally, the result holds if all eigenvalues of lie in for all .[6]

Completely solvable Lie algebras

A Lie algebra is called completely solvable or split solvable if it has an elementary sequence 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 .[7]

Examples

Then is solvable, but not split solvable.[7] It is isomorphic with the Lie algebra of the group of translations and rotations in the plane.

Solvable Lie groups

Because the term "solvable" is also used for solvable groups in group theory, there are several possible definitions of solvable Lie group. For a Lie group , there is

See also

Notes

References

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