Inmathematics, aLie algebra${\displaystyle \mathfrak{g}}$ issolvable if its derived series terminates in the zero subalgebra. Thederived Lie algebra of the Lie algebra${\displaystyle \mathfrak{g}}$ is the subalgebra of${\displaystyle \mathfrak{g}}$, denoted

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

that consists of all linear combinations ofLie brackets of pairs of elements of${\displaystyle \mathfrak{g}}$. Thederived series is the sequence of subalgebras

${\displaystyle \mathfrak{g}\supseteq [\mathfrak{g},\mathfrak{g}]\supseteq [[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]\supseteq [[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]],[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]]\supseteq ...}$

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 thederived series forcommutator subgroups ingroup theory, and solvable Lie algebras are analogs ofsolvable groups.

Anynilpotent Lie algebra isa fortiori solvable but the converse is not true. The solvable Lie algebras and thesemisimple Lie algebras form two large and generally complementary classes, as is shown by theLevi decomposition. The solvable Lie algebras are precisely those that can be obtained fromsemidirect products, starting from 0 and adding one dimension at a time.^[2]

A maximal solvable subalgebra is called aBorel subalgebra. The largest solvableideal of a Lie algebra is called theradical.

Let${\displaystyle \mathfrak{g}}$ be a finite-dimensional Lie algebra over a field ofcharacteristic0. The following are equivalent.

• (i)${\displaystyle \mathfrak{g}}$ is solvable.
• (ii)${\displaystyle \mathrm{a}\mathrm{d}(\mathfrak{g})}$, theadjoint representation of${\displaystyle \mathfrak{g}}$, is solvable.
• (iii) There is a finite sequence of ideals${\displaystyle {\mathfrak{a}}_i}$ of${\displaystyle \mathfrak{g}}$:

  ${\displaystyle \mathfrak{g}={\mathfrak{a}}_0\supset {\mathfrak{a}}_1\supset ...{\mathfrak{a}}_r=0,[{\mathfrak{a}}_i,{\mathfrak{a}}_i]\subset {\mathfrak{a}}_{i+1}\mathrm{\forall }i.}$

• (iv)${\displaystyle [\mathfrak{g},\mathfrak{g}]}$ is nilpotent.^[3]
• (v) For${\displaystyle \mathfrak{g}}$${\displaystyle n}$-dimensional, there is a finite sequence of subalgebras${\displaystyle {\mathfrak{a}}_i}$ of${\displaystyle \mathfrak{g}}$:

  ${\displaystyle \mathfrak{g}={\mathfrak{a}}_0\supset {\mathfrak{a}}_1\supset ...{\mathfrak{a}}_n=0,\dim {\mathfrak{a}}_i/{\mathfrak{a}}_{i+1}=1\mathrm{\forall }i,}$

with each${\displaystyle {\mathfrak{a}}_{i+1}}$ an ideal in${\displaystyle {\mathfrak{a}}_i}$.^[4] A sequence of this type is called anelementary sequence.

• (vi) There is a finite sequence of subalgebras${\displaystyle {\mathfrak{g}}_i}$ of${\displaystyle \mathfrak{g}}$,

  ${\displaystyle \mathfrak{g}={\mathfrak{g}}_0\supset {\mathfrak{g}}_1\supset ...{\mathfrak{g}}_r=0,}$

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

• (vii) TheKilling form${\displaystyle B}$ of${\displaystyle \mathfrak{g}}$ satisfies${\displaystyle B(X,Y)=0}$ for allX in${\displaystyle \mathfrak{g}}$ andY in${\displaystyle [\mathfrak{g},\mathfrak{g}]}$.^[6] This isCartan's criterion for solvability.

Lie's Theorem states that if${\displaystyle V}$ is a finite-dimensional vector space over an algebraically closed field ofcharacteristic zero, and${\displaystyle \mathfrak{g}}$ is a solvable Lie algebra, and if${\displaystyle \pi }$ is arepresentation of${\displaystyle \mathfrak{g}}$ over${\displaystyle V}$, then there exists a simultaneouseigenvector${\displaystyle v\in V}$ of the endomorphisms${\displaystyle \pi (X)}$ for all elements${\displaystyle X\in \mathfrak{g}}$.^[7]

• Every Lie subalgebra and quotient of a solvable Lie algebra are solvable.^[8]
• Given a Lie algebra${\displaystyle \mathfrak{g}}$ and an ideal${\displaystyle \mathfrak{h}}$ in it,

  ${\displaystyle \mathfrak{g}}$ is solvable if and only if both${\displaystyle \mathfrak{h}}$ and${\displaystyle \mathfrak{g}/\mathfrak{h}}$ are solvable.^[8][2]

The analogous statement is true for nilpotent Lie algebras provided${\displaystyle \mathfrak{h}}$ is contained in the center. Thus, an extension of a solvable algebra by a solvable algebra is solvable, while acentral 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${\displaystyle \mathfrak{a},\mathfrak{b}\subset \mathfrak{g}}$ are solvable ideals, then so is${\displaystyle \mathfrak{a}+\mathfrak{b}}$.^[1] Consequently, if${\displaystyle \mathfrak{g}}$ is finite-dimensional, then there is a unique solvable ideal${\displaystyle \mathfrak{r}\subset \mathfrak{g}}$ containing all solvable ideals in${\displaystyle \mathfrak{g}}$. This ideal is theradical of${\displaystyle \mathfrak{g}}$.^[2]
• A solvable Lie algebra${\displaystyle \mathfrak{g}}$ has a unique largest nilpotent ideal${\displaystyle \mathfrak{n}}$, called thenilradical, the set of all${\displaystyle X\in \mathfrak{g}}$ such that${\displaystyle {\mathrm{a}\mathrm{d}}_X}$ is nilpotent. IfD is any derivation of${\displaystyle \mathfrak{g}}$, then${\displaystyle D(\mathfrak{g})\subset \mathfrak{n}}$.^[9]

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

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

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

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

Another class of examples comes fromnilpotent Lie algebras since the adjoint representation is solvable. Some examples include the upper-diagonal matrices, such as the class of matrices of the form

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

and is denoted${\displaystyle {\mathfrak{b}}_k}$.

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

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

Asemisimple Lie algebra${\displaystyle \mathfrak{l}}$ is never solvable since itsradical${\displaystyle \text{Rad}(\mathfrak{l})}$, which is the largest solvable ideal in${\displaystyle \mathfrak{l}}$, is trivial.^{[1]page 11}

Because the term "solvable" is also used forsolvable groups ingroup theory, there are several possible definitions ofsolvable Lie group. For aLie group${\displaystyle G}$, there is

• termination of the usualderived series of the group${\displaystyle G}$ (as an abstract group);
• termination of the closures of the derived series;
• having a solvable Lie algebra

• Cartan's criterion
• Killing form
• Lie-Kolchin theorem
• Solvmanifold
• Dixmier mapping

• Fulton, W.;Harris, J. (1991).Representation theory. A first course. Graduate Texts in Mathematics. Vol. 129. New York: Springer-Verlag.ISBN 978-0-387-97527-6.MR 1153249.
• Humphreys, James E. (1972).Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. Vol. 9. New York: Springer-Verlag.ISBN 0-387-90053-5.
• Knapp, A. W. (2002).Lie groups beyond an introduction. Progress in Mathematics. Vol. 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser.ISBN 0-8176-4259-5..
• Serre, Jean-Pierre (2001).Complex Semisimple Lie Algebras. Berlin: Springer.ISBN 3-5406-7827-1.

• EoM articleLie algebra, solvable
• EoM articleLie group, solvable

• Properties of Lie algebras

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