Let ${\displaystyle {\mathfrak {g}}}$  be a finite-dimensional Lie algebra over a field of characteristic 0. The following are equivalent.

• (i) ${\displaystyle {\mathfrak {g}}}$  is solvable.
• (ii) ${\displaystyle {\rm {ad}}({\mathfrak {g}})}$ , the adjoint representation of ${\displaystyle {\mathfrak {g}}}$ , is solvable.
• (iii) There is a finite sequence of ideals ${\displaystyle {\mathfrak {a}}_{i}}$  of ${\displaystyle {\mathfrak {g}}}$ :

  ${\displaystyle {\mathfrak {g}}={\mathfrak {a}}_{0}\supset {\mathfrak {a}}_{1}\supset ...{\mathfrak {a}}_{r}=0,\quad [{\mathfrak {a}}_{i},{\mathfrak {a}}_{i}]\subset {\mathfrak {a}}_{i+1}\,\,\forall i.}$ 

• (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 ${\displaystyle {\mathfrak {g}}}$ :

  ${\displaystyle {\mathfrak {g}}={\mathfrak {a}}_{0}\supset {\mathfrak {a}}_{1}\supset ...{\mathfrak {a}}_{n}=0,\quad \operatorname {dim} {\mathfrak {a}}_{i}/{\mathfrak {a}}_{i+1}=1\,\,\forall i,}$ 

with each ${\displaystyle {\mathfrak {a}}_{i+1}}$  an ideal in ${\displaystyle {\mathfrak {a}}_{i}}$ .^[4] A sequence of this type is called an elementary 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) The Killing form ${\displaystyle B}$  of ${\displaystyle {\mathfrak {g}}}$  satisfies ${\displaystyle B(X,Y)=0}$  for all X in ${\displaystyle {\mathfrak {g}}}$  and Y in ${\displaystyle [{\mathfrak {g}},{\mathfrak {g}}]}$ .^[6] This is Cartan's criterion for solvability.

Lie's Theorem states that if ${\displaystyle V}$  is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and ${\displaystyle {\mathfrak {g}}}$  is a solvable Lie algebra, and if ${\displaystyle \pi }$  is a representation of ${\displaystyle {\mathfrak {g}}}$  over ${\displaystyle V}$ , then there exists a simultaneous eigenvector ${\displaystyle v\in V}$  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 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 ${\displaystyle {\mathfrak {a}},{\mathfrak {b}}\subset {\mathfrak {g}}}$  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 the radical of ${\displaystyle {\mathfrak {g}}}$ .^[2]
• A solvable Lie algebra ${\displaystyle {\mathfrak {g}}}$  has a unique largest nilpotent ideal ${\displaystyle {\mathfrak {n}}}$ , called the nilradical, the set of all ${\displaystyle X\in {\mathfrak {g}}}$  such that ${\displaystyle {\rm {ad}}_{X}}$  is nilpotent. If D is any derivation of ${\displaystyle {\mathfrak {g}}}$ , then ${\displaystyle D({\mathfrak {g}})\subset {\mathfrak {n}}}$ .^[9]

A Lie algebra ${\displaystyle {\mathfrak {g}}}$  is called completely solvable or split solvable if it has an elementary sequence{(V) As above definition} 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 ${\displaystyle {\rm {ad}}_{X}}$  are in ${\displaystyle k}$  for all ${\displaystyle X}$  in ${\displaystyle {\mathfrak {g}}}$ .^[2]

Abelian Lie algebras edit

Every abelian 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 ${\displaystyle {\mathfrak {gl}}(n)}$ , which are of the form

${\displaystyle \left\{{\begin{bmatrix}*&0&0\\0&*&0\\0&0&*\end{bmatrix}}\right\}}$ 

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.

Nilpotent Lie algebras edit

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 form

${\displaystyle \left\{{\begin{bmatrix}0&*&*\\0&0&*\\0&0&0\end{bmatrix}}\right\}}$ 

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

${\displaystyle \left\{{\begin{bmatrix}*&*&*\\0&*&*\\0&0&*\end{bmatrix}}\right\}}$ 

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

Solvable but not split-solvable edit

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

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

Then ${\displaystyle {\mathfrak {g}}}$  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 edit

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

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

• termination of the usual derived 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

• EoM article Lie algebra, solvable
• EoM article Lie group, solvable

• 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..
• Jean-Pierre Serre: Complex Semisimple Lie Algebras, Springer, Berlin, 2001. ISBN 3-5406-7827-1