# 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.^{[1]} 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*.

## Contents

## Characterizations

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

- (i)
**g**is solvable. - (ii) ad(
**g**), the adjoint representation of**g**, is solvable. - (iii) There is a finite sequence of ideals
**a**_{i}of**g**: - (iv) [
**g**,**g**] is nilpotent. - (v) For
**g***n*-dimensional, there is a finite sequence of subalgebras**a**_{i}of**g**:

- with each
**a**_{i + 1}an ideal in**a**_{i}.^{[2]}A sequence of this type is called an**elementary sequence**.

- such that
**g**_{i + 1}is an ideal in**g**_{i}and**g**_{i}/**g**_{i + 1}is abelian.^{[3]}

- (vii)
**g**is solvable if and only if its Killing form*B*satisfies*B*(*X*,*Y*) = 0 for all Template:Mvar in**g**and and Template:Mvar in [**g**,**g**].^{[4]}This is Cartan's criterion for solvability.

## Properties

Lie's Theorem states that if *V* is a finite-dimensional vector space over an algebraically closed field **K** of characteristic zero, and **g** is a solvable linear Lie algebra over a subfield **k** of **K**, and if *π* is a **representation** of **g** over *V*, then there exists a simultaneous eigenvector *v* ∈ *V* of the matrices *π*(*X*) for all elements *X* ∈ **g**. More generally, the result holds if all eigenvalues of *π*(*X*) lie in **K** for all *X* ∈ **g**.^{[5]}

- Every Lie subalgebra, quotient and extension of a solvable Lie algebra is solvable.
- A solvable nonzero Lie algebra has a nonzero abelian ideal, the last nonzero term in the derived series.
^{[6]} - A homomorphic image of a solvable Lie algebra is solvable.
^{[6]} - If
**a**is a solvable ideal in**g**and**g**/**a**is solvable, then**g**is solvable.^{[6]} - If
**g**is finite-dimensional, then there is a unique solvable ideal**r**⊂**g**containing all solvable ideals in**g**. This ideal is the**radical**of**g**, denoted rad**g**.^{[6]} - If
**a**,**b**⊂**g**are solvable ideals, then so is**a**+**b**.^{[1]} - A solvable Lie algebra
**g**has a unique largest nilpotent ideal**n**, the set of all*X*∈**g**such that ad_{X}is nilpotent. If Template:Mvar is any derivation of**g**, then*D*(**g**) ⊂**n**.^{[7]}

## Completely solvable Lie algebras

A Lie algebra **g** is called **completely solvable** or **split solvable** if it has a elementary sequence of ideals in **g** from 0 to **g**. A finite-dimensional nilpotent Lie algebra is completely solvable, and a completely solvable Lie algebra is solvable. Over an algebraically closed field and solvable Lie algebra is completely solvable, but the 3-dimensional real Lie algebra of the group of Euclidean isometries of the plane is solvable but not completely solvable.

- (a) A solvable Lie algebra
**g**is split solvable if and only if the eigenvalues of ad_{X}are in**k**for all*X*in**g**.^{[6]}

## Examples

- A semisimple Lie algebra is
*not*solvable.^{[1]} - Every abelian Lie algebra is solvable.
- Every nilpotent Lie algebra is solvable.
- Let
**b**_{k}be a subalgebra of**gl**_{k}consisting of upper triangular matrices. Then**b**_{k}is solvable. - Let
**g**be the the set of matrices on the form

- Then
**g**is solvable, but not split solvable.^{[6]}It is isomorphic with the Lie algebra of the group of translations and rotations in the plane.

## Solvable Lie groups

The terminology arises from the solvable groups of abstract group theory. There are several possible definitions of **solvable Lie group**. For a Lie group *G*, there is

- termination of the usual derived series, in other words taking
*G*as an abstract group; - termination of the closures of the derived series;
- having a solvable Lie algebra.

To have equivalence one needs to assume *G* connected. For connected Lie groups, these definitions are the same, and the derived series of Lie algebras are the Lie algebra of the derived series of (closed) subgroups.

## See also

## External links

## Notes

- ↑
^{1.0}^{1.1}^{1.2}Template:Harvnb - ↑ Template:Harvnb Proposition 1.23.
- ↑ Template:Harvnb
- ↑ Template:Harvnb Proposition 1.46.
- ↑ Template:Harvnb Theorem 1.25.
- ↑
^{6.0}^{6.1}^{6.2}^{6.3}^{6.4}^{6.5}Template:Harvnb - ↑ Template:Harvnb Proposition 1.40.

## References

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