Solvable Lie algebra explained

ak{g}

is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie algebra

ak{g}

is the subalgebra of

ak{g}

, denoted

[ak{g},ak{g}]

that consists of all linear combinations of Lie brackets of pairs of elements of

ak{g}

. 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

ak{g}

be a finite-dimensional Lie algebra over a field of characteristic . The following are equivalent.

ak{g}

is solvable.

{\rmad}(ak{g})

, the adjoint representation of

ak{g}

, is solvable.

ak{a}i

of

ak{g}

:

ak{g}=ak{a}0\supsetak{a}1\supset...ak{a}r=0,   [ak{a}i,ak{a}i]\subsetak{a}i+1\foralli.

[ak{g},ak{g}]

is nilpotent.[1]

ak{g}

n

-dimensional, there is a finite sequence of subalgebras

ak{a}i

of

ak{g}

:

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

ak{a}i+1

an ideal in

ak{a}i

.[2] A sequence of this type is called an elementary sequence.

ak{g}i

of

ak{g}

,

ak{g}=ak{g}0\supsetak{g}1\supset...ak{g}r=0,

such that

ak{g}i+1

is an ideal in

ak{g}i

and

ak{g}i/ak{g}i+1

is abelian.

B

of

ak{g}

satisfies

B(X,Y)=0

for all in

ak{g}

and in

[ak{g},ak{g}]

.[3] 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

ak{g}

is a solvable Lie algebra, and if

\pi

is a representation of

ak{g}

over

V

, then there exists a simultaneous eigenvector

v\inV

of the endomorphisms

\pi(X)

for all elements

X\inak{g}

.[4]

akg

and an ideal

akh

in it,

ak{g}

is solvable if and only if both

akh

and

ak{g}/akh

are solvable.

The analogous statement is true for nilpotent Lie algebras provided

akh

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.

ak{a},ak{b}\subak{g}

are solvable ideals, then so is

ak{a}+ak{b}

. Consequently, if

ak{g}

is finite-dimensional, then there is a unique solvable ideal

ak{r}\subak{g}

containing all solvable ideals in

ak{g}

. This ideal is the radical of

ak{g}

.

ak{g}

has a unique largest nilpotent ideal

ak{n}

, called the nilradical, the set of all

X\inak{g}

such that

{\rmad}X

is nilpotent. If is any derivation of

ak{g}

, then

D(ak{g})\subak{n}

.[5]

Completely solvable Lie algebras

A Lie algebra

ak{g}

is called completely solvable or split solvable if it has an elementary sequence of ideals in

ak{g}

from

0

to

ak{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

ak{g}

is split solvable if and only if the eigenvalues of

{\rmad}X

are in

k

for all

X

in

ak{g}

.

Examples

Abelian Lie algebras

ak{a}

is solvable by definition, since its commutator

[ak{a},ak{a}]=0

. This includes the Lie algebra of diagonal matrices in

ak{gl}(n)

, which are of the form

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

for

n=3

. The Lie algebra structure on a vector space

V

given by the trivial bracket

[m,n]=0

for any two matrices

m,n\inEnd(V)

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

ak{gl}(n)

form a solvable Lie algebra. This includes matrices of the form

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

and is denoted

ak{b}k

.

Solvable but not split-solvable

Let

ak{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),   \theta,x,y\inR.

Then

ak{g}

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

ak{l}

is never solvable since its radical

Rad(ak{l})

, which is the largest solvable ideal in

ak{l}

, is trivial. page 11

Solvable Lie groups

G

, there is

G

(as an abstract group);

See also

References

External links

Notes and References

  1. Proposition 1.39.
  2. Proposition 1.23.
  3. Proposition 1.46.
  4. Theorem 1.25.
  5. Proposition 1.40.

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