Editing Reductive Lie algebra

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In [[mathematics]], a [[Lie algebra]] is '''reductive''' if its [[Adjoint endomorphism|adjoint representation]] is [[Lie algebra representation|completely reducible]], hence the name. More concretely, a Lie algebra is reductive if it is a [[direct sum of Lie algebras|direct sum]] of a [[semisimple Lie algebra]] and an [[abelian Lie algebra]]: <math>\mathfrak{g} = \mathfrak{s} \oplus \mathfrak{a};</math> there are alternative characterizations, given below.

== Examples ==
The most basic example is the Lie algebra <math>\mathfrak{gl}_n</math> of <math>n \times n</math> matrices with the commutator as Lie bracket, or more abstractly as the [[endomorphism]] algebra of an ''n''-dimensional [[vector space]], <math>\mathfrak{gl}(V).</math> This is the Lie algebra of the [[general linear group]] GL(''n''), and is reductive as it decomposes as <math>\mathfrak{gl}_n = \mathfrak{sl}_n \oplus \mathfrak{k},</math> corresponding to [[traceless]] matrices and [[scalar matrices]].

Any [[semisimple Lie algebra]] or [[abelian Lie algebra]] is ''a fortiori'' reductive.

Over the real numbers, [[compact Lie algebra]]s are reductive.

== Definitions ==
A Lie algebra <math>\mathfrak{g}</math> over a field of characteristic 0 is called reductive if any of the following equivalent conditions are satisfied:
# The [[Adjoint representation of a Lie group|adjoint representation]] (the action by bracketing) of <math>\mathfrak{g}</math> is [[Semisimple Lie algebra|completely reducible]] (a [[direct sum of representations|direct sum]] of irreducible representations).
# <math>\mathfrak{g}</math> admits a faithful, completely reducible, finite-dimensional representation.
# The [[Radical of a Lie algebra|radical]] of <math>\mathfrak{g}</math> equals the center: <math>\mathfrak{r}(\mathfrak{g}) = \mathfrak{z}(\mathfrak{g}).</math>
#:The radical always contains the center, but need not equal it.
# <math>\mathfrak{g}</math> is the direct sum of a semisimple ideal <math>\mathfrak{s}_0</math> and its center <math>\mathfrak{z}(\mathfrak{g}): </math> <math>\mathfrak{g} = \mathfrak{s}_0 \oplus \mathfrak{z}(\mathfrak{g}).</math>
#:Compare to the [[Levi decomposition]], which decomposes a Lie algebra as its radical (which is solvable, not abelian in general) and a Levi subalgebra (which is semisimple).
# <math>\mathfrak{g}</math> is a direct sum of a semisimple Lie algebra <math>\mathfrak{s}</math> and an abelian Lie algebra <math>\mathfrak{a}</math>: <math>\mathfrak{g} = \mathfrak{s} \oplus \mathfrak{a}.</math>
# <math>\mathfrak{g}</math> is a direct sum of prime ideals: <math>\mathfrak{g} = \textstyle{\sum \mathfrak{g}_i}.</math>
Some of these equivalences are easily seen. For example, the center and radical of <math>\mathfrak{s} \oplus \mathfrak{a}</math> is <math>\mathfrak{a},</math> while if the radical equals the center the Levi decomposition yields a decomposition <math>\mathfrak{g} = \mathfrak{s}_0 \oplus \mathfrak{z}(\mathfrak{g}).</math> Further, simple Lie algebras and the trivial 1-dimensional Lie algebra <math>\mathfrak{k}</math> are prime ideals.

== Properties ==
Reductive Lie algebras are a generalization of semisimple Lie algebras, and share many properties with them: many properties of semisimple Lie algebras depend only on the fact that they are reductive. Notably, the [[unitarian trick]] of [[Hermann Weyl]] works for reductive Lie algebras.

The associated [[reductive Lie group]]s are of significant interest: the [[Langlands program]] is based on the premise that what is done for one reductive Lie group should be done for all.{{clarify|date=May 2013}}

The intersection of reductive Lie algebras and solvable Lie algebras is exactly abelian Lie algebras (contrast with the intersection of semisimple and solvable Lie algebras being trivial).

==References==
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== External links ==
* ''[https://encyclopediaofmath.org/wiki/Lie_algebra,_reductive Lie algebra, reductive],'' A.L. Onishchik, in ''Encyclopaedia of Mathematics,'' {{ISBN|1-4020-0609-8}}, SpringerLink

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[[Category:Properties of Lie algebras]]