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In [[algebra]], the '''nilradical''' of a [[Lie algebra]] is a nilpotent [[Ideal (Lie algebra)|ideal]], which is as large as possible.  
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The nilradical <math>\mathfrak{nil}(\mathfrak g)</math> of a finite dimensional Lie algebra <math>\mathfrak{g}</math> is its maximal [[nilpotent ideal]], which exists because the sum of any two nilpotent ideals is nilpotent. It is an ideal in the [[radical of a Lie algebra|radical]] <math>\mathfrak{rad}(\mathfrak{g})</math> of the Lie algebra <math>\mathfrak{g}</math>. The quotient of a Lie algebra by its nilradical is a [[reductive Lie algebra|reductive]] Lie algebra <math>\mathfrak{g}^{\mathrm{red}}</math>. However, the corresponding [[short exact sequence]]
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:<math> 0 \to \mathfrak{nil}(\mathfrak g)\to \mathfrak g\to \mathfrak{g}^{\mathrm{red}}\to 0</math>
does not split in general (i.e., there isn't always a ''subalgebra'' complementary to <math>\mathfrak{nil}(\mathfrak g)</math> in <math>\mathfrak{g}</math>). This is in contrast to the [[Levi decomposition]]: the short exact sequence
:<math> 0 \to \mathfrak{rad}(\mathfrak g)\to \mathfrak g\to \mathfrak{g}^{\mathrm{ss}}\to 0</math>
does split (essentially because the quotient <math>\mathfrak{g}^{\mathrm{ss}}</math> is semisimple).
 
==See also==
* [[Levi decomposition]]
* [[Nilradical of a ring]], a notion in ring theory.
 
==References==
 
* {{Fulton-Harris}}
* {{citation|title=Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras|first1=Arkadi L.|last1=Onishchik|author1-link=Arkadi L. Onishchik|first2=Ėrnest Borisovich|last2= Vinberg|author2-link=Ernest Borisovich Vinberg |publisher=Springer|year= 1994|isbn=978-3-540-54683-2}}.
 
[[Category:Lie algebras]]

Latest revision as of 12:19, 29 December 2014

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