# Solvable Lie algebra

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In mathematics, a type of algebra

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In [mathematics](/source/Mathematics), a [Lie algebra](/source/Lie_algebra) g {\displaystyle {\mathfrak {g}}} is **solvable** if its derived series terminates in the zero subalgebra. The *derived Lie algebra* of the Lie algebra g {\displaystyle {\mathfrak {g}}} is the subalgebra of g {\displaystyle {\mathfrak {g}}} , denoted

- [ g , g ] {\displaystyle [{\mathfrak {g}},{\mathfrak {g}}]}

that consists of all linear combinations of [Lie brackets](/source/Lie_bracket) of pairs of elements of g {\displaystyle {\mathfrak {g}}} . The *derived series* is the sequence of subalgebras

- g ⊇ [ g , g ] ⊇ [ [ g , g ] , [ g , g ] ] ⊇ [ [ [ g , g ] , [ g , g ] ] , [ [ g , g ] , [ g , g ] ] ] ⊇ . . . {\displaystyle {\mathfrak {g}}\supseteq [{\mathfrak {g}},{\mathfrak {g}}]\supseteq [[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]\supseteq [[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]],[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]]\supseteq ...}

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](/source/Derived_series) for [commutator subgroups](/source/Commutator_subgroup) in [group theory](/source/Group_theory), and solvable Lie algebras are analogs of [solvable groups](/source/Solvable_group).

Any [nilpotent Lie algebra](/source/Nilpotent_Lie_algebra) is [a fortiori](/source/A_fortiori) solvable but the converse is not true. The solvable Lie algebras and the [semisimple Lie algebras](/source/Semisimple_Lie_algebra) form two large and generally complementary classes, as is shown by the [Levi decomposition](/source/Levi_decomposition). The solvable Lie algebras are precisely those that can be obtained from [semidirect products](/source/Lie_algebra_extension), starting from 0 and adding one dimension at a time.[2]

A maximal solvable subalgebra is called a [Borel subalgebra](/source/Borel_subalgebra). The largest solvable [ideal](/source/Ideal_(Lie_algebra)) of a Lie algebra is called the [radical](/source/Radical_of_Lie_algebra).

## Characterizations

Let g {\displaystyle {\mathfrak {g}}} be a finite-dimensional Lie algebra over a field of [characteristic](/source/Characteristic_(algebra)) 0. The following are equivalent.

- (i) g {\displaystyle {\mathfrak {g}}} is solvable.

- (ii) a d ( g ) {\displaystyle {\rm {ad}}({\mathfrak {g}})} , the [adjoint representation](/source/Adjoint_representation_of_a_Lie_algebra) of g {\displaystyle {\mathfrak {g}}} , is solvable.

- (iii) There is a finite sequence of ideals a i {\displaystyle {\mathfrak {a}}_{i}} of g {\displaystyle {\mathfrak {g}}} : - g = a 0 ⊃ a 1 ⊃ . . . a r = 0 , [ a i , a i ] ⊂ a i + 1 ∀ i . {\displaystyle {\mathfrak {g}}={\mathfrak {a}}_{0}\supset {\mathfrak {a}}_{1}\supset ...{\mathfrak {a}}_{r}=0,\quad [{\mathfrak {a}}_{i},{\mathfrak {a}}_{i}]\subset {\mathfrak {a}}_{i+1}\,\,\forall i.}

- (iv) [ g , g ] {\displaystyle [{\mathfrak {g}},{\mathfrak {g}}]} is nilpotent.[3]

- (v) For g {\displaystyle {\mathfrak {g}}} n {\displaystyle n} -dimensional, there is a finite sequence of subalgebras a i {\displaystyle {\mathfrak {a}}_{i}} of g {\displaystyle {\mathfrak {g}}} : - g = a 0 ⊃ a 1 ⊃ . . . a n = 0 , dim ⁡ a i / a i + 1 = 1 ∀ i , {\displaystyle {\mathfrak {g}}={\mathfrak {a}}_{0}\supset {\mathfrak {a}}_{1}\supset ...{\mathfrak {a}}_{n}=0,\quad \operatorname {dim} {\mathfrak {a}}_{i}/{\mathfrak {a}}_{i+1}=1\,\,\forall i,}

- with each a i + 1 {\displaystyle {\mathfrak {a}}_{i+1}} an ideal in a i {\displaystyle {\mathfrak {a}}_{i}} .[4] A sequence of this type is called an **elementary sequence**.

- (vi) There is a finite sequence of subalgebras g i {\displaystyle {\mathfrak {g}}_{i}} of g {\displaystyle {\mathfrak {g}}} , - g = g 0 ⊃ g 1 ⊃ . . . g r = 0 , {\displaystyle {\mathfrak {g}}={\mathfrak {g}}_{0}\supset {\mathfrak {g}}_{1}\supset ...{\mathfrak {g}}_{r}=0,}

- such that g i + 1 {\displaystyle {\mathfrak {g}}_{i+1}} is an ideal in g i {\displaystyle {\mathfrak {g}}_{i}} and g i / g i + 1 {\displaystyle {\mathfrak {g}}_{i}/{\mathfrak {g}}_{i+1}} is abelian.[5]

- (vii) The [Killing form](/source/Killing_form) B {\displaystyle B} of g {\displaystyle {\mathfrak {g}}} satisfies B ( X , Y ) = 0 {\displaystyle B(X,Y)=0} for all X in g {\displaystyle {\mathfrak {g}}} and Y in [ g , g ] {\displaystyle [{\mathfrak {g}},{\mathfrak {g}}]} .[6] This is [Cartan's criterion for solvability](/source/Cartan's_criterion#Cartan's_criterion_for_solvability).

## Properties

[Lie's Theorem](/source/Lie's_Theorem) states that if V {\displaystyle V} is a finite-dimensional vector space over an algebraically closed field of [characteristic zero](/source/Characteristic_(field_theory)), and g {\displaystyle {\mathfrak {g}}} is a solvable Lie algebra, and if π {\displaystyle \pi } is a [representation](/source/Lie_algebra_representation) of g {\displaystyle {\mathfrak {g}}} over V {\displaystyle V} , then there exists a simultaneous [eigenvector](/source/Eigenvector) v ∈ V {\displaystyle v\in V} of the endomorphisms π ( X ) {\displaystyle \pi (X)} for all elements X ∈ g {\displaystyle X\in {\mathfrak {g}}} .[7]

- Every Lie subalgebra and quotient of a solvable Lie algebra are solvable.[8]

- Given a Lie algebra g {\displaystyle {\mathfrak {g}}} and an ideal h {\displaystyle {\mathfrak {h}}} in it, - g {\displaystyle {\mathfrak {g}}} is solvable if and only if both h {\displaystyle {\mathfrak {h}}} and g / h {\displaystyle {\mathfrak {g}}/{\mathfrak {h}}} are solvable.[8][2]

- The analogous statement is true for nilpotent Lie algebras provided h {\displaystyle {\mathfrak {h}}} 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.[2]

- If a , b ⊂ g {\displaystyle {\mathfrak {a}},{\mathfrak {b}}\subset {\mathfrak {g}}} are solvable ideals, then so is a + b {\displaystyle {\mathfrak {a}}+{\mathfrak {b}}} .[1] Consequently, if g {\displaystyle {\mathfrak {g}}} is finite-dimensional, then there is a unique solvable ideal r ⊂ g {\displaystyle {\mathfrak {r}}\subset {\mathfrak {g}}} containing all solvable ideals in g {\displaystyle {\mathfrak {g}}} . This ideal is the **[radical](/source/Radical_of_a_Lie_algebra)** of g {\displaystyle {\mathfrak {g}}} .[2]

- A solvable Lie algebra g {\displaystyle {\mathfrak {g}}} has a unique largest nilpotent ideal n {\displaystyle {\mathfrak {n}}} , called the [nilradical](/source/Nilradical_of_a_Lie_algebra), the set of all X ∈ g {\displaystyle X\in {\mathfrak {g}}} such that a d X {\displaystyle {\rm {ad}}_{X}} is nilpotent. If D is any derivation of g {\displaystyle {\mathfrak {g}}} , then D ( g ) ⊂ n {\displaystyle D({\mathfrak {g}})\subset {\mathfrak {n}}} .[9]

## Completely solvable Lie algebras

A Lie algebra g {\displaystyle {\mathfrak {g}}} is called **completely solvable** or **split solvable** if it has an elementary sequence of ideals in g {\displaystyle {\mathfrak {g}}} from 0 {\displaystyle 0} to g {\displaystyle {\mathfrak {g}}} . 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 3 {\displaystyle 3} -dimensional real Lie algebra of the group of Euclidean isometries of the plane is solvable but not completely solvable.

A solvable Lie algebra g {\displaystyle {\mathfrak {g}}} is split solvable if and only if the eigenvalues of a d X {\displaystyle {\rm {ad}}_{X}} are in k {\displaystyle k} for all X {\displaystyle X} in g {\displaystyle {\mathfrak {g}}} .[2]

## Examples

### Abelian Lie algebras

Every [abelian Lie algebra](/source/Abelian_Lie_algebra) a {\displaystyle {\mathfrak {a}}} is solvable by definition, since its commutator [ a , a ] = 0 {\displaystyle [{\mathfrak {a}},{\mathfrak {a}}]=0} . This includes the Lie algebra of diagonal matrices in g l ( n ) {\displaystyle {\mathfrak {gl}}(n)} , which are of the form

{ [ ∗ 0 0 0 ∗ 0 0 0 ∗ ] } {\displaystyle \left\{{\begin{bmatrix}*&0&0\\0&*&0\\0&0&*\end{bmatrix}}\right\}}

for n = 3 {\displaystyle n=3} . The Lie algebra structure on a vector space V {\displaystyle V} given by the trivial bracket [ m , n ] = 0 {\displaystyle [m,n]=0} for any two matrices m , n ∈ End ( V ) {\displaystyle m,n\in {\text{End}}(V)} gives another example.

### Nilpotent Lie algebras

Another class of examples comes from [nilpotent Lie algebras](/source/Nilpotent_Lie_algebra) since the adjoint representation is solvable. Some examples include the upper-diagonal matrices, such as the class of matrices of the form

{ [ 0 ∗ ∗ 0 0 ∗ 0 0 0 ] } {\displaystyle \left\{{\begin{bmatrix}0&*&*\\0&0&*\\0&0&0\end{bmatrix}}\right\}}

called the Lie algebra of **strictly upper triangular matrices**. In addition, the Lie algebra of **upper diagonal matrices** in g l ( n ) {\displaystyle {\mathfrak {gl}}(n)} form a solvable Lie algebra. This includes matrices of the form

{ [ ∗ ∗ ∗ 0 ∗ ∗ 0 0 ∗ ] } {\displaystyle \left\{{\begin{bmatrix}*&*&*\\0&*&*\\0&0&*\end{bmatrix}}\right\}}

and is denoted b k {\displaystyle {\mathfrak {b}}_{k}} .

### Solvable but not split-solvable

Let g {\displaystyle {\mathfrak {g}}} be the set of matrices on the form

X = ( 0 θ x − θ 0 y 0 0 0 ) , θ , x , y ∈ R . {\displaystyle X=\left({\begin{matrix}0&\theta &x\\-\theta &0&y\\0&0&0\end{matrix}}\right),\quad \theta ,x,y\in \mathbb {R} .}

Then g {\displaystyle {\mathfrak {g}}} is solvable, but not split solvable.[2] It is isomorphic with the Lie algebra of the group of translations and rotations in the plane.

### Non-example

A [semisimple Lie algebra](/source/Semisimple_Lie_algebra) l {\displaystyle {\mathfrak {l}}} is never solvable since its [radical](/source/Radical_of_a_Lie_algebra) Rad ( l ) {\displaystyle {\text{Rad}}({\mathfrak {l}})} , which is the largest solvable ideal in l {\displaystyle {\mathfrak {l}}} , is trivial.[1] page 11

## Solvable Lie groups

Because the term "solvable" is also used for [solvable groups](/source/Solvable_group) in [group theory](/source/Group_theory), there are several possible definitions of **solvable Lie group**. For a [Lie group](/source/Lie_group) G {\displaystyle G} , there is

- termination of the usual [derived series](/source/Derived_series) of the group G {\displaystyle G} (as an abstract group);

- termination of the closures of the derived series;

- having a solvable Lie algebra

## See also

- [Cartan's criterion](/source/Cartan's_criterion)

- [Killing form](/source/Killing_form)

- [Lie–Kolchin theorem](/source/Lie%E2%80%93Kolchin_theorem)

- [Solvmanifold](/source/Solvmanifold)

- [Dixmier mapping](/source/Dixmier_mapping)

## Notes

1. ^ [***a***](#cite_ref-Humphreys_1_1-0) [***b***](#cite_ref-Humphreys_1_1-1) [***c***](#cite_ref-Humphreys_1_1-2) [Humphreys 1972](#CITEREFHumphreys1972)

1. ^ [***a***](#cite_ref-Knapp_1_2-0) [***b***](#cite_ref-Knapp_1_2-1) [***c***](#cite_ref-Knapp_1_2-2) [***d***](#cite_ref-Knapp_1_2-3) [***e***](#cite_ref-Knapp_1_2-4) [***f***](#cite_ref-Knapp_1_2-5) [Knapp 2002](#CITEREFKnapp2002)

1. **[^](#cite_ref-3)** [Knapp 2002](#CITEREFKnapp2002) Proposition 1.39.

1. **[^](#cite_ref-4)** [Knapp 2002](#CITEREFKnapp2002) Proposition 1.23.

1. **[^](#cite_ref-Fulton_1_5-0)** [Fulton & Harris 1991](#CITEREFFultonHarris1991)

1. **[^](#cite_ref-6)** [Knapp 2002](#CITEREFKnapp2002) Proposition 1.46.

1. **[^](#cite_ref-7)** [Knapp 2002](#CITEREFKnapp2002) Theorem 1.25.

1. ^ [***a***](#cite_ref-Serre_def_8-0) [***b***](#cite_ref-Serre_def_8-1) [Serre 2001](#CITEREFSerre2001), Ch. I, § 6, Definition 2.

1. **[^](#cite_ref-9)** [Knapp 2002](#CITEREFKnapp2002) Proposition 1.40.

## References

- [Fulton, W.](/source/William_Fulton_(mathematician)); [Harris, J.](/source/Joe_Harris_(mathematician)) (1991). *Representation theory. A first course*. Graduate Texts in Mathematics. Vol. 129. New York: Springer-Verlag. [ISBN](/source/ISBN_(identifier)) [978-0-387-97527-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-97527-6). [MR](/source/MR_(identifier)) [1153249](https://mathscinet.ams.org/mathscinet-getitem?mr=1153249).

- Humphreys, James E. (1972). [*Introduction to Lie Algebras and Representation Theory*](https://archive.org/details/introductiontoli00jame). Graduate Texts in Mathematics. Vol. 9. New York: Springer-Verlag. [ISBN](/source/ISBN_(identifier)) [0-387-90053-5](https://en.wikipedia.org/wiki/Special:BookSources/0-387-90053-5).

- [Knapp, A. W.](/source/A._W._Knapp) (2002). *Lie groups beyond an introduction*. Progress in Mathematics. Vol. 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. [ISBN](/source/ISBN_(identifier)) [0-8176-4259-5](https://en.wikipedia.org/wiki/Special:BookSources/0-8176-4259-5)..

- Serre, Jean-Pierre (2001). *Complex Semisimple Lie Algebras*. Berlin: Springer. [ISBN](/source/ISBN_(identifier)) [3-5406-7827-1](https://en.wikipedia.org/wiki/Special:BookSources/3-5406-7827-1).

## External links

- [EoM article *Lie algebra, solvable*](https://encyclopediaofmath.org/wiki/Lie_algebra,_solvable)

- [EoM article *Lie group, solvable*](https://encyclopediaofmath.org/wiki/Lie_group,_solvable)

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