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| In [[Lie theory]] and [[representation theory]], the '''Levi decomposition''', conjectured by Killing and Cartan and proved by {{harvs|txt|authorlink=Eugenio Elia Levi | first=Eugenio Elia|last= Levi|year=1905}}, states that any finite dimensional real [[Lie algebra]] ''g'' is the semidirect product of a [[solvable Lie algebra|solvable]] ideal and a [[semisimple Lie algebra|semisimple]] subalgebra.
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| One is its '''radical''', a maximal solvable ideal, and the other is a semisimple subalgebra, called a '''Levi subalgebra'''. Levi decomposition implies that any finite dimensional Lie algebra is a [[semidirect product]] of a solvable Lie algebra and a semisimple Lie algebra.
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| When viewed as a factor-algebra of ''g'', this semisimple Lie algebra is also called the '''Levi factor''' of ''g''.
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| Moreover, [[Malcev]] (1942) showed that any two Levi subalgebras are [[conjugacy|conjugate]] by an (inner) automorphism of the form
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| :<math>\exp(\mathrm{ad}(z))\ </math>
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| where ''z'' is in the [[Nilradical of a Lie algebra|nilradical]] ('''Levi–Malcev theorem''').
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| ==Application==
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| To a certain extent, the decomposition can be used to reduce problems about finite dimensional Lie algebras and Lie groups to separate problems about Lie algebras in these two special classes, solvable and semisimple.
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| ==Extensions of the results==
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| In representation theory, Levi decomposition of [[parabolic subgroup]]s of a reductive group is needed to construct a large family of the so-called [[parabolic induction|parabolically induced]] representations. The [[Langlands decomposition]] is a slight refinement of the Levi decomposition for parabolic subgroups used in this context.
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| Analogous statements hold for simply connected [[Lie group]]s, and, as shown by [[George Mostow]], for algebraic Lie algebras and simply connected [[algebraic group]]s over a field of [[characteristic (algebra)|characteristic]] zero.
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| There is no analogue of the Levi decomposition for most infinite-dimensional Lie algebras; for example [[affine Lie algebra]]s have a radical consisting of their center, but cannot be written as a semidirect product of the center and another Lie algebra. The Levi decomposition also fails for finite dimensional algebras over fields of positive characteristic.
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| == See also ==
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| *[[Lie group decompositions]]
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| ==References==
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| *Jacobson, ''Lie algebras''
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| *{{Citation
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| | last = Levi
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| | first = Eugenio Elia
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| | author-link =
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| | title = Sulla struttura dei gruppi finiti e continui
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| | journal = Atti della Reale Accademia delle Scienze di Torino.
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| | language = [[Italian language|Italian]]
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| | volume = XL
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| | date =
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| | year = 1905
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| | url = http://www.archive.org/details/attidellarealeac40real
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| | archiveurl = http://www.archive.org/stream/attidellarealeac40real#page/550/mode/2up
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| | archivedate = March 05, 2009
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| | pages = 551–565
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| | id =
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| | jfm = 36.0217.02
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| }} Reprinted in: Opere Vol. 1, Edizione Cremonese, Rome (1959), p. 101.
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| *{{Citation | last1=Maltsev | first1=Anatoly I. | authorlink=Anatoly Maltsev | title=On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra | id= | year=1942 | journal=C. R. (Doklady) Acad. Sci. URSS (N.S.) | volume=36 | pages=42–45| mr= 0007397 | zbl= 0060.08004 }}.
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| ==External links==
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| *{{springer|id=Levi%E2%80%93Mal%27tsev_decomposition|title=Levi-Mal'tsev decomposition|author=A.I. Shtern}}
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| [[Category:Lie algebras]]
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