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| {{Lie groups}}
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| In [[mathematics]], a [[Lie algebra]] <math>\mathfrak{g}</math> is '''solvable''' if its ''[[derived series]]'' terminates in the zero subalgebra. That is, writing
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| :<math>[\mathfrak{g},\mathfrak{g}]</math> | |
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| for the '''derived Lie algebra''' of <math>\mathfrak{g}</math>, generated by the set of values
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| :[''x'',''y'']
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| for ''x'' and ''y'' in <math>\mathfrak{g}</math>, the derived series
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| :<math> \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 ...</math>
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| becomes constant eventually at 0.
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| Any [[nilpotent Lie algebra]] is solvable, ''[[a fortiori]]'', 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]].
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| A maximal solvable subalgebra is called a ''[[Borel subalgebra]]''. The largest solvable [[ideal (Lie algebra)|ideal]] is called the ''[[Radical of Lie algebra|radical]]''.
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| == Properties == | |
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| Let <math>\mathfrak{g}</math> be a finite dimensional Lie algebra over a field of [[Characteristic_(algebra)|characteristic]] 0. The following are equivalent.
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| *(i) <math>\mathfrak{g}</math> is solvable.
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| *(ii) <math>\operatorname{ad}(\mathfrak{g})</math>, the [[adjoint representation of a Lie algebra|adjoint representation]] of <math>\mathfrak{g}</math>, is solvable.
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| *(iii) There is a finite sequence of ideals <math>\mathfrak{a}_i</math> of <math>\mathfrak{g}</math> such that:
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| *:<math>\mathfrak{g} = \mathfrak{a}_0 \supset \mathfrak{a}_1 \supset ... \mathfrak{a}_r = 0</math> where <math>[\mathfrak{a}_i, \mathfrak{a}_i] \subset \mathfrak{a}_{i+1}</math> for all <math>i</math>.
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| *(iv) <math>[\mathfrak{g}, \mathfrak{g}]</math> is nilpotent.
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| [[Lie's Theorem]] states that if <math>V</math> is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and <math>\mathfrak{g}</math> is a solvable linear Lie algebra over <math>V</math>, then there exists a basis of <math>V</math> relative to which the matrices of all elements of <math>\mathfrak{g}</math> are upper triangular.
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| ==Completely solvable Lie algebras==
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| A Lie algebra <math>\mathfrak{g}</math> is called completely solvable if it has a finite chain of ideals from 0 to <math>\mathfrak{g}</math> such that each has codimension 1 in the next. A finite-dimensional nilpotent Lie algebra is completely solvable, and a completely solvable Lie algebra is solvable. Over an algebraically closed field and 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.
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| == Example ==
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| * Every [[abelian Lie algebra]] is solvable.
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| * Every [[nilpotent Lie algebra]] is solvable.
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| * Every Lie subalgebra, quotient and extension of a solvable [[Lie algebra]] is solvable.
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| * Let <math>\mathfrak{b}_k</math> be a subalgebra of <math>\mathfrak{gl}_k</math> consisting of upper triangular matrices. Then <math>\mathfrak{b}_k</math> is solvable.
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| ==Solvable Lie groups==
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| The terminology arises from the [[solvable group]]s of abstract [[group theory]]. There are several possible definitions of '''solvable Lie group'''. For a [[Lie group]] ''G'', there is
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| * termination of the usual [[derived series]], in other words taking ''G'' as an abstract group;
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| * termination of the closures of the derived series;
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| * having a solvable Lie algebra.
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| To have equivalence one needs to assume ''G'' connected. For connected Lie groups, these definitions are the same, and the derived series of Lie algebras are the Lie algebra of the derived series of (closed) subgroups.
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| ==See also==
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| *[[Cartan's criterion]]
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| *[[Killing form]]
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| *[[Lie-Kolchin theorem]]
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| *[[Solvmanifold]]
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| *[[Dixmier mapping]]
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| ==External links==
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| *[http://eom.springer.de/l/l058520.htm EoM article ''Lie algebra, solvable'']
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| *[http://eom.springer.de/l/l058690.htm EoM article ''Lie group, solvable'']
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| ==References== | |
| * Humphreys, James E. ''Introduction to Lie Algebras and Representation Theory''. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1972. ISBN 0-387-90053-5
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| [[Category:Properties of Lie algebras]]
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