Solvable Lie algebra

In mathematics, a Lie algebra ${\mathfrak {g}}$ is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie algebra ${\mathfrak {g}}$ is the subalgebra of ${\mathfrak {g}}$, denoted

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

that consists of all linear combinations of Lie brackets of pairs of elements of ${\mathfrak {g}}$. The derived series is the sequence of subalgebras

${\mathfrak {g}}\geq [{\mathfrak {g}},{\mathfrak {g}}]\geq [[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]\geq [[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]],[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]]\geq ...$

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.

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.

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

Characterizations

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

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

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

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

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

with each ${\mathfrak {a}}_{i+1}$ an ideal in ${\mathfrak {a}}_{i}$. A sequence of this type is called an elementary sequence.

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

  ${\mathfrak {g}}={\mathfrak {g}}_{0}\supset {\mathfrak {g}}_{1}\supset ...{\mathfrak {g}}_{r}=0,$

such that ${\mathfrak {g}}_{i+1}$ is an ideal in ${\mathfrak {g}}_{i}$ and ${\mathfrak {g}}_{i}/{\mathfrak {g}}_{i+1}$ is abelian.

• (vii) The Killing form $B$ of ${\mathfrak {g}}$ satisfies $B(X,Y)=0$ for all X in ${\mathfrak {g}}$ and Y in $[{\mathfrak {g}},{\mathfrak {g}}]$. 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 of characteristic zero, and ${\mathfrak {g}}$ is a solvable Lie algebra, and if $\pi$ is a representation of ${\mathfrak {g}}$ over $V$, then there exists a simultaneous eigenvector $v\in V$ of the endomorphisms $\pi (X)$ for all elements $X\in {\mathfrak {g}}$.

• Every Lie subalgebra and quotient of a solvable Lie algebra is solvable.
• Given a Lie algebra ${\mathfrak {g}}$ and an ideal ${\mathfrak {h}}$,

  ${\mathfrak {g}}$ is solvable if and only if both ${\mathfrak {h}}$ and ${\mathfrak {g}}/{\mathfrak {h}}$ are solvable.

Note that the analogous statement is true for nilpotent Lie algebras provided ${\mathfrak {h}}$ 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.
• If ${\mathfrak {a}}$ is a solvable ideal in ${\mathfrak {g}}$ and ${\mathfrak {g}}/{\mathfrak {a}}$ is solvable, then ${\mathfrak {g}}$ is solvable.
• If ${\mathfrak {g}}$ is finite-dimensional, then there is a unique solvable ideal ${\mathfrak {r}}\subset {\mathfrak {g}}$ containing all solvable ideals in ${\mathfrak {g}}$. This ideal is the radical of ${\mathfrak {g}}$, denoted ${\rm {rad}}{\mathfrak {g}}$.
• If ${\mathfrak {a}},{\mathfrak {b}}\subset {\mathfrak {g}}$ are solvable ideals, then so is ${\mathfrak {a}}+{\mathfrak {b}}$.
• A solvable Lie algebra ${\mathfrak {g}}$ has a unique largest nilpotent ideal ${\mathfrak {n}}$, the set of all $X\in {\mathfrak {g}}$ such that ${\rm {ad}}_{X}$ is nilpotent. If D is any derivation of ${\mathfrak {g}}$, then $D({\mathfrak {g}})\subset {\mathfrak {n}}$.

Completely solvable Lie algebras

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

A solvable Lie algebra ${\mathfrak {g}}$ is split solvable if and only if the eigenvalues of ${\rm {ad}}_{X}$ are in $k$ for all $X$ in ${\mathfrak {g}}$.

Examples

• A semisimple Lie algebra is never solvable.
• Every abelian Lie algebra is solvable.
• Every nilpotent Lie algebra is solvable.
• Let ${\mathfrak {b}}_{k}$ be the subalgebra of ${\rm {gl}}_{k}$ consisting of upper triangular matrices. Then ${\mathfrak {b}}_{k}$ is solvable.
• Let ${\mathfrak {g}}$ be the set of matrices on the form

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

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

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 $G$, there is

• termination of the usual derived series of the group $G$ (as an abstract group);
• termination of the closures of the derived series;
• having a solvable Lie algebra

See also

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

External links

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

References

• 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. {{cite book}}: Invalid |ref=harv (help)
• 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. {{cite book}}: Invalid |ref=harv (help)
• 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. {{cite book}}: Invalid |ref=harv (help).
• Jean-Pierre Serre: Complex Semisimple Lie Algebras, Springer, Berlin, 2001. ISBN 3-5406-7827-1