Solvable Lie algebra

In [[mathematics]], a [[Lie algebra]] <math>\mathfrak{g}</math> is '''solvable''' if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra <math>\mathfrak{g}</math> is the subalgebra of <math>\mathfrak{g}</math>, denoted

:<math>[\mathfrak{g},\mathfrak{g}]</math>

that consists of all linear combinations of [[Lie bracket]]s of pairs of elements of <math>\mathfrak{g}</math>. The ''derived series'' is the sequence of subalgebras

:<math> \mathfrak{g} \supseteq [\mathfrak{g},\mathfrak{g}] \supseteq [[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]] \supseteq [[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]],[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]] \supseteq ...</math>

If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is called solvable.<ref name=Humphreys_1>{{harvnb|Humphreys|1972}}</ref> The derived series for Lie algebras is analogous to the [[derived series]] for [[commutator subgroup]]s in [[group theory]], and solvable Lie algebras are analogs of [[solvable group]]s.

Any [[nilpotent Lie algebra]] is [[a fortiori]] solvable but the converse is not true. The solvable Lie algebras and the [[semisimple Lie algebra]]s 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.<ref name="Knapp_1"/>

A maximal solvable subalgebra is called a [[Borel subalgebra]]. The largest solvable [[ideal (Lie algebra)|ideal]] of a Lie algebra is called the [[Radical of Lie algebra|radical]].

== Characterizations == Let <math>\mathfrak{g}</math> be a finite-dimensional Lie algebra over a field of [[Characteristic (algebra)|characteristic]] {{math|0}}. The following are equivalent. *(i) <math>\mathfrak{g}</math> is solvable. *(ii) <math>{\rm ad}(\mathfrak{g})</math>, the [[adjoint representation of a Lie algebra|adjoint representation]] of <math>\mathfrak{g}</math>, is solvable. *(iii) There is a finite sequence of ideals <math>\mathfrak{a}_i</math> of <math>\mathfrak{g}</math>: *:<math>\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 .</math> *(iv) <math>[\mathfrak{g}, \mathfrak{g}]</math> is nilpotent.<ref>{{harvnb|Knapp|2002}} Proposition 1.39.</ref> *(v) For <math>\mathfrak{g}</math> <math>n</math>-dimensional, there is a finite sequence of subalgebras <math>\mathfrak{a}_i</math> of <math>\mathfrak{g}</math>: *:<math>\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,</math> :with each <math>\mathfrak{a}_{i+1}</math> an ideal in <math>\mathfrak{a}_i</math>.<ref>{{harvnb|Knapp|2002}} Proposition 1.23.</ref> A sequence of this type is called an '''elementary sequence'''. *(vi) There is a finite sequence of subalgebras <math>\mathfrak{g}_i</math> of <math>\mathfrak{g}</math>, *:<math>\mathfrak{g} = \mathfrak{g}_0 \supset \mathfrak{g}_1 \supset ... \mathfrak{g}_r = 0,</math> :such that <math>\mathfrak{g}_{i+1}</math> is an ideal in <math>\mathfrak{g}_i</math> and <math>\mathfrak{g}_i/\mathfrak{g}_{i+1}</math> is abelian.<ref name= Fulton_1>{{harvnb|Fulton|Harris|1991}}</ref> *(vii) The [[Killing form]] <math>B</math> of <math>\mathfrak{g}</math> satisfies <math>B(X,Y)=0</math> for all {{mvar|X}} in <math>\mathfrak{g}</math> and {{mvar|Y}} in <math>[\mathfrak{g}, \mathfrak{g}]</math>.<ref>{{harvnb|Knapp|2002}} Proposition 1.46.</ref> This is [[Cartan's criterion#Cartan's criterion for solvability|Cartan's criterion for solvability]].

== Properties == [[Lie's Theorem]] states that if <math>V</math> is a finite-dimensional vector space over an algebraically closed field of [[Characteristic (field theory)|characteristic zero]], and <math>\mathfrak{g}</math> is a solvable Lie algebra, and if <math>\pi</math> is a [[Lie algebra representation|representation]] of <math>\mathfrak{g}</math> over <math>V</math>, then there exists a simultaneous [[eigenvector]] <math>v \in V</math> of the endomorphisms <math>\pi(X)</math> for all elements <math>X \in \mathfrak{g}</math>.<ref>{{harvnb|Knapp|2002}} Theorem 1.25.</ref>

*Every Lie subalgebra and quotient of a solvable Lie algebra are solvable.<ref name="Serre def">{{harvnb|Serre|2001|loc=Ch. I, § 6, Definition 2.}}</ref> *Given a Lie algebra <math>\mathfrak g</math> and an ideal <math>\mathfrak h</math> in it, *:<math>\mathfrak{g}</math> is solvable if and only if both <math>\mathfrak h</math> and <math>\mathfrak{g}/\mathfrak h</math> are solvable.<ref name="Serre def" /><ref name=Knapp_1/> :The analogous statement is true for nilpotent Lie algebras provided <math>\mathfrak h</math> 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.<ref name=Knapp_1>{{harvnb|Knapp|2002}}</ref> * If <math>\mathfrak{a}, \mathfrak{b} \sub \mathfrak{g}</math> are solvable ideals, then so is <math>\mathfrak{a} + \mathfrak{b}</math>.<ref name=Humphreys_1/> Consequently, if <math>\mathfrak{g}</math> is finite-dimensional, then there is a unique solvable ideal <math>\mathfrak{r} \sub \mathfrak{g}</math> containing all solvable ideals in <math>\mathfrak{g}</math>. This ideal is the '''[[radical of a Lie algebra|radical]]''' of <math>\mathfrak{g}</math>.<ref name=Knapp_1/> *A solvable Lie algebra <math>\mathfrak{g}</math> has a unique largest nilpotent ideal <math>\mathfrak{n}</math>, called the [[Nilradical of a Lie algebra|nilradical]], the set of all <math>X \in \mathfrak{g}</math> such that <math>{\rm ad}_X</math> is nilpotent. If {{mvar|D}} is any derivation of <math>\mathfrak{g}</math>, then <math>D(\mathfrak{g}) \sub \mathfrak{n}</math>.<ref>{{harvnb|Knapp|2002}} Proposition 1.40.</ref>

==Completely solvable Lie algebras==

A Lie algebra <math>\mathfrak{g}</math> is called '''completely solvable''' or '''split solvable''' if it has an elementary sequence of ideals in <math>\mathfrak{g}</math> from <math>0</math> to <math>\mathfrak{g}</math>. 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 <math>3</math>-dimensional real Lie algebra of the group of Euclidean isometries of the plane is solvable but not completely solvable.

A solvable Lie algebra <math>\mathfrak{g}</math> is split solvable if and only if the eigenvalues of <math>{\rm ad}_X</math> are in <math>k</math> for all <math>X</math> in <math>\mathfrak{g}</math>.<ref name=Knapp_1/>

== Examples ==

=== Abelian Lie algebras === Every [[abelian Lie algebra]] <math>\mathfrak{a}</math> is solvable by definition, since its commutator <math>[\mathfrak{a},\mathfrak{a}] = 0</math>. This includes the Lie algebra of diagonal matrices in <math>\mathfrak{gl}(n)</math>, which are of the form<blockquote><math>\left\{ \begin{bmatrix} * & 0 & 0 \\ 0 & * & 0 \\ 0 & 0 & * \end{bmatrix} \right\}</math></blockquote>for <math>n = 3</math>. The Lie algebra structure on a vector space <math>V</math> given by the trivial bracket <math>[m,n] = 0</math> for any two matrices <math>m,n \in \text{End}(V)</math> gives another example.

=== Nilpotent Lie algebras === Another class of examples comes from [[nilpotent Lie algebra]]s since the adjoint representation is solvable. Some examples include the upper-diagonal matrices, such as the class of matrices of the form<blockquote><math>\left\{ \begin{bmatrix} 0 & * & * \\ 0 & 0 & * \\ 0 & 0 & 0 \end{bmatrix} \right\}</math></blockquote>called the Lie algebra of '''strictly upper triangular matrices'''. In addition, the Lie algebra of '''upper diagonal matrices''' in <math>\mathfrak{gl}(n)</math> form a solvable Lie algebra. This includes matrices of the form<blockquote><math>\left\{ \begin{bmatrix} * & * & * \\ 0 & * & * \\ 0 & 0 & * \end{bmatrix} \right\}</math></blockquote>and is denoted <math>\mathfrak{b}_k</math>.

=== Solvable but not split-solvable === Let <math>\mathfrak{g}</math> be the set of matrices on the form<blockquote><math>X = \left(\begin{matrix}0 & \theta & x\\ -\theta & 0 & y\\ 0 & 0 & 0\end{matrix}\right), \quad \theta, x, y \in \mathbb{R}.</math></blockquote>Then <math>\mathfrak{g}</math> is solvable, but not split solvable.<ref name="Knapp_1" /> It is isomorphic with the Lie algebra of the group of translations and rotations in the plane.

=== Non-example === A [[semisimple Lie algebra]] <math>\mathfrak{l}</math> is never solvable since its [[Radical of a Lie algebra|radical]] <math>\text{Rad}(\mathfrak{l})</math>, which is the largest solvable ideal in <math>\mathfrak{l}</math>, is trivial.<ref name="Humphreys_1" /> <sup>page 11</sup>

==Solvable Lie groups==

Because the term "solvable" is also used for [[solvable group]]s in [[group theory]], there are several possible definitions of '''solvable Lie group'''. For a [[Lie group]] <math>G</math>, there is

* termination of the usual [[derived series]] of the group <math>G</math> (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]]

==Notes== {{Reflist}}

==References== *{{cite book|last1=Fulton|last2=Harris|first1=W.|first2=J.|year=1991|publisher=Springer-Verlag|location=New York|series=Graduate Texts in Mathematics|volume=129|isbn=978-0-387-97527-6|mr=1153249|authorlink1=William Fulton (mathematician)|authorlink2=Joe Harris (mathematician)|title=Representation theory. A first course}} *{{cite book|last=Humphreys|first=James E.|title=Introduction to Lie Algebras and Representation Theory|series=Graduate Texts in Mathematics|volume=9|publisher=Springer-Verlag|location=New York|year=1972|isbn=0-387-90053-5|url-access=registration|url=https://archive.org/details/introductiontoli00jame}} *{{cite book|author-link=A. W. Knapp|last=Knapp|first=A. W.|title=Lie groups beyond an introduction|isbn=0-8176-4259-5|publisher=Birkhäuser|series=Progress in Mathematics|volume=120|edition=2nd|year=2002|location=Boston·Basel·Berlin}}. *{{cite book|first=Jean-Pierre|last=Serre|title=Complex Semisimple Lie Algebras|publisher=Springer|location=Berlin|year=2001|isbn=3-5406-7827-1}}

==External links== *[https://encyclopediaofmath.org/wiki/Lie_algebra,_solvable EoM article ''Lie algebra, solvable''] *[https://encyclopediaofmath.org/wiki/Lie_group,_solvable EoM article ''Lie group, solvable'']

[[Category:Properties of Lie algebras]]