Solvable Lie algebra explained
is
solvable if its derived series terminates in the zero subalgebra. The
derived Lie algebra of the Lie algebra
is the subalgebra of
, denoted
that consists of all linear combinations of Lie brackets of pairs of elements of
. The
derived series is the sequence of subalgebras
ak{g}\geq[ak{g},ak{g}]\geq[[ak{g},ak{g}],[ak{g},ak{g}]]\geq[[[ak{g},ak{g}],[ak{g},ak{g}]],[[ak{g},ak{g}],[ak{g},ak{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, and solvable Lie algebras are analogs of solvable groups.
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. The solvable Lie algebras are precisely those that can be obtained from semidirect products, starting from 0 and adding one dimension at a time.
A maximal solvable subalgebra is called a Borel subalgebra. The largest solvable ideal of a Lie algebra is called the radical.
Characterizations
Let
be a finite-dimensional Lie algebra over a field of
characteristic . The following are equivalent.
is solvable.
, the adjoint representation of
, is solvable.
- (iii) There is a finite sequence of ideals
of
:
ak{g}=ak{a}0\supsetak{a}1\supset...ak{a}r=0, [ak{a}i,ak{a}i]\subsetak{a}i+1\foralli.
is nilpotent.
[1]
-dimensional, there is a finite sequence of subalgebras
of
:
ak{g}=ak{a}0\supsetak{a}1\supset...ak{a}n=0, \operatorname{dim}ak{a}i/ak{a}i=1\foralli,
with each
an ideal in
.
[2] A sequence of this type is called an
elementary sequence.
- (vi) There is a finite sequence of subalgebras
of
,
ak{g}=ak{g}0\supsetak{g}1\supset...ak{g}r=0,
such that
is an ideal in
and
is abelian.
of
satisfies
for all in
and in
.
[3] This is Cartan's criterion for solvability.
Properties
Lie's Theorem states that if
is a finite-dimensional vector space over an algebraically closed field of
characteristic zero, and
is a solvable Lie algebra, and if
is a
representation of
over
, then there exists a simultaneous
eigenvector
of the endomorphisms
for all elements
.
[4] - Every Lie subalgebra and quotient of a solvable Lie algebra are solvable.
- Given a Lie algebra
and an ideal
in it,
is solvable if and only if both
and
are solvable.
The analogous statement is true for nilpotent Lie algebras provided
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
are solvable ideals, then so is
. Consequently, if
is finite-dimensional, then there is a unique solvable ideal
containing all solvable ideals in
. This ideal is the
radical of
.
has a unique largest nilpotent ideal
, called the
nilradical, the set of all
such that
is nilpotent. If is any derivation of
, then
.
[5] Completely solvable Lie algebras
A Lie algebra
is called
completely solvable or
split solvable if it has an elementary sequence of ideals in
from
to
. 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
-dimensional real Lie algebra of the group of Euclidean isometries of the plane is solvable but not completely solvable.
A solvable Lie algebra
is split solvable if and only if the eigenvalues of
are in
for all
in
.
Examples
Abelian Lie algebras
is solvable by definition, since its commutator
. This includes the Lie algebra of diagonal matrices in
, which are of the form
\left\{\begin{bmatrix}
*&0&0\\
0&*&0\\
0&0&*
\end{bmatrix}\right\}
for
. The Lie algebra structure on a vector space
given by the trivial bracket
for any two matrices
gives another example.
Nilpotent Lie algebras
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
\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
form a solvable Lie algebra. This includes matrices of the form
\left\{\begin{bmatrix}
*&*&*\\
0&*&*\\
0&0&*
\end{bmatrix}\right\}
and is denoted
.
Solvable but not split-solvable
Let
be the set of matrices on the form
X=\left(\begin{matrix}0&\theta&x\ -\theta&0&y\ 0&0&0\end{matrix}\right), \theta,x,y\inR.
Then
is solvable, but not split solvable. It is isomorphic with the Lie algebra of the group of translations and rotations in the plane.
Non-example
is never solvable since its
radical
, which is the largest solvable ideal in
, is trivial.
page 11Solvable Lie groups
, there is
- termination of the usual derived series of the group
(as an abstract group);
- termination of the closures of the derived series;
- having a solvable Lie algebra
See also
References
- Book: Fulton. Harris. W.. J.. 1991. Springer-Verlag. New York. Graduate Texts in Mathematics. 129. 978-0-387-97527-6. 1153249. William Fulton (mathematician). Joe Harris (mathematician). Representation theory. A first course.
- Book: Humphreys, James E.. Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. 9. Springer-Verlag. New York. 1972. 0-387-90053-5. registration.
- Book: Knapp, A. W.. A. W. Knapp. Lie groups beyond an introduction. 0-8176-4259-5. Birkhäuser. Progress in Mathematics. 120. 2nd. 2002. Boston·Basel·Berlin. .
- Book: Serre, Jean-Pierre. Complex Semisimple Lie Algebras. Springer. Berlin. 2001. 3-5406-7827-1.
External links
Notes and References
- Proposition 1.39.
- Proposition 1.23.
- Proposition 1.46.
- Theorem 1.25.
- Proposition 1.40.