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Wikipedia / Solvable Lie algebra

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

that consists of all linear combinations of Lie brackets. 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.

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

Characterizations

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

Properties

Lie's Theorem is a finite-dimensional vector space over an algebraically closed field of characteristic zero is a representation, then there exists a simultaneous eigenvector.

  • Every Lie subalgebra and quotient of a solvable Lie algebra are solvable.

  • in it,

    1. are solvable.

    is contained in the center. Thus, an extension of a solvable algebra by a solvable algebra is solvable, while a central extension of a nilpotent algebra by a nilpotent algebra is nilpotent.

  • A solvable nonzero Lie algebra has a nonzero abelian ideal, the last nonzero term in the derived series.

  • .. This ideal is the radical.

  • , called the nilradical is nilpotent. If .

Completely solvable Lie algebras

is called completely solvable or split solvable-dimensional real Lie algebra of the group of Euclidean isometries of the plane is solvable but not completely solvable.

.

Examples

Abelian Lie algebras

Every abelian Lie algebra, which are of the form gives another example.

Nilpotent Lie algebras

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 form a solvable Lie algebra. This includes matrices of the form.

Solvable but not split-solvable

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

Non-example

A semisimple Lie algebra is never solvable since its radical, is trivial. page 11

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

  • termination of the usual derived series (as an abstract group);

  • termination of the closures of the derived series;

  • having a solvable Lie algebra

See also

Notes

References

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