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In mathematics, a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
\mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted : mathfrak,\mathfrak/math> that consists of all linear combinations of
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s of pairs of elements of \mathfrak. The ''derived series'' is the sequence of subalgebras : \mathfrak \geq mathfrak,\mathfrak\geq \mathfrak,\mathfrak mathfrak,\mathfrak \geq [ \mathfrak,\mathfrak mathfrak,\mathfrak, \mathfrak,\mathfrak mathfrak,\mathfrak] \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 In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest norma ...
for
commutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
s in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, and solvable Lie algebras are analogs of
solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group (mathematics), group that can be constructed from abelian groups using Group extension, extensions. Equivalently, a solvable group is a ...
s. Any
nilpotent Lie algebra In mathematics, a Lie algebra \mathfrak is nilpotent if its lower central series terminates in the zero subalgebra. The ''lower central series'' is the sequence of subalgebras : \mathfrak \geq mathfrak,\mathfrak\geq mathfrak, mathfrak,\mathfra ...
is a fortiori solvable but the converse is not true. The solvable Lie algebras and the semisimple Lie algebra">a fortiori">mathfrak,\mathfra ...
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 Lie algebra extension, semidirect products, starting from 0 and adding one dimension at a time. A maximal solvable subalgebra is called a
Borel subalgebra In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie group, ...
. The largest solvable
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of a Lie algebra is called the
radical Radical may refer to: Politics and ideology Politics * Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe an ...
.


Characterizations

Let \mathfrak be a finite-dimensional Lie algebra over a field of characteristic . The following are equivalent. *(i) \mathfrak is solvable. *(ii) (\mathfrak), the
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL( ...
of \mathfrak, is solvable. *(iii) There is a finite sequence of ideals \mathfrak_i of \mathfrak: *:\mathfrak = \mathfrak_0 \supset \mathfrak_1 \supset ... \mathfrak_r = 0, \quad mathfrak_i, \mathfrak_i\subset \mathfrak_ \,\, \forall i . *(iv) mathfrak, \mathfrak/math> is nilpotent. *(v) For \mathfrak n-dimensional, there is a finite sequence of subalgebras \mathfrak_i of \mathfrak: *:\mathfrak = \mathfrak_0 \supset \mathfrak_1 \supset ... \mathfrak_n = 0, \quad \operatorname \mathfrak_/\mathfrak_ = 1 \,\, \forall i, :with each \mathfrak_ an ideal in \mathfrak_i. A sequence of this type is called an elementary sequence. *(vi) There is a finite sequence of subalgebras \mathfrak_i of \mathfrak, *:\mathfrak = \mathfrak_0 \supset \mathfrak_1 \supset ... \mathfrak_r = 0, :such that \mathfrak_ is an ideal in \mathfrak_i and \mathfrak_i/\mathfrak_ is abelian. *(vii) The
Killing form In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) s ...
B of \mathfrak satisfies B(X,Y)=0 for all in \mathfrak and in mathfrak, \mathfrak/math>. This is Cartan's criterion for solvability.


Properties

Lie's Theorem In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if \pi: \mathfrak \to \mathfrak(V) is a finite-dimensional representation of a solvable Lie algebra, then ...
states that if V is a finite-dimensional vector space over an algebraically closed field of
characteristic zero In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
, and \mathfrak is a solvable Lie algebra, and if \pi is a
representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
of \mathfrak over V, then there exists a simultaneous
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
v \in V of the endomorphisms \pi(X) for all elements X \in \mathfrak. *Every Lie subalgebra and quotient of a solvable Lie algebra are solvable. *Given a Lie algebra \mathfrak g and an ideal \mathfrak h in it, *:\mathfrak is solvable if and only if both \mathfrak h and \mathfrak/\mathfrak h are solvable. :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, \mathfrak \sub \mathfrak are solvable ideals, then so is \mathfrak + \mathfrak. Consequently, if \mathfrak is finite-dimensional, then there is a unique solvable ideal \mathfrak \sub \mathfrak containing all solvable ideals in \mathfrak. This ideal is the
radical Radical may refer to: Politics and ideology Politics * Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe an ...
of \mathfrak. *A solvable Lie algebra \mathfrak has a unique largest nilpotent ideal \mathfrak, called the nilradical, the set of all X \in \mathfrak such that _X is nilpotent. If is any derivation of \mathfrak, then D(\mathfrak) \sub \mathfrak. Proposition 1.40.


Completely solvable Lie algebras

A Lie algebra \mathfrak is called completely solvable or split solvable if it has an elementary sequence of ideals in \mathfrak from 0 to \mathfrak. 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 is split solvable if and only if the eigenvalues of _X are in k for all X in \mathfrak.


Examples


Abelian Lie algebras

Every
abelian Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
\mathfrak is solvable by definition, since its commutator mathfrak,\mathfrak= 0. This includes the Lie algebra of diagonal matrices in \mathfrak(n), which are of the form
\left\
for n = 3. The Lie algebra structure on a vector space V given by the trivial bracket ,n= 0 for any two matrices m,n \in \text(V) gives another example.


Nilpotent Lie algebras

Another class of examples comes from
nilpotent Lie algebra In mathematics, a Lie algebra \mathfrak is nilpotent if its lower central series terminates in the zero subalgebra. The ''lower central series'' is the sequence of subalgebras : \mathfrak \geq mathfrak,\mathfrak\geq mathfrak,[\mathfrak,\mathfra ...
s since the adjoint representation is solvable. Some examples include the upper-diagonal matrices, such as the class of matrices of the form
\left\
called the Lie algebra of strictly upper triangular matrices. In addition, the Lie algebra of upper diagonal matrices in \mathfrak(n) form a solvable Lie algebra. This includes matrices of the form
\left\
and is denoted \mathfrak_k.


Solvable but not split-solvable

Let \mathfrak be the set of matrices on the form
X = \left(\begin0 & \theta & x\\ -\theta & 0 & y\\ 0 & 0 & 0\end\right), \quad \theta, x, y \in \mathbb.
Then \mathfrak 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

A semisimple Lie algebra \mathfrak is never solvable since its Radical of a Lie algebra, radical \text(\mathfrak), which is the largest solvable ideal in \mathfrak, is trivial. page 11


Solvable Lie groups

Because the term "solvable" is also used for
solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group (mathematics), group that can be constructed from abelian groups using Group extension, extensions. Equivalently, a solvable group is a ...
s in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, there are several possible definitions of solvable Lie group. For a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
G, there is * termination of the usual
derived series In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest norma ...
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 In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) s ...
* Lie-Kolchin theorem *
Solvmanifold In mathematics, a solvmanifold is a homogeneous space of a connected solvable Lie group. It may also be characterized as a quotient of a connected solvable Lie group by a closed subgroup. (Some authors also require that the Lie group be simply-co ...
*
Dixmier mapping In mathematics, the Dixmier mapping describes the space Prim(''U''(''g'')) of primitive ideals of the universal enveloping algebra ''U''(''g'') of a finite-dimensional solvable Lie algebra ''g'' over an algebraically closed field of characteristic ...


Notes


External links


EoM article ''Lie algebra, solvable''EoM article ''Lie group, solvable''


References

* * *. *Jean-Pierre Serre: Complex Semisimple Lie Algebras, Springer, Berlin, 2001. {{Authority control Properties of Lie algebras