Editing Lie group decomposition

{{onesource|date=September 2009}}
In [[mathematics]], '''Lie group decompositions''' are used to analyse the structure of [[Lie group]]s and associated objects, by showing how they are built up out of [[subgroup]]s. They are essential technical tools in the [[representation theory]] of Lie groups and [[Lie algebra]]s; they can also be used to study the [[algebraic topology]] of such groups and associated [[homogeneous space]]s. Since the use of Lie group methods became one of the standard techniques in twentieth century mathematics, many phenomena can now be referred back to decompositions.

The same ideas are often applied to Lie groups, Lie algebras, [[algebraic group]]s and [[p-adic number]] analogues, making it harder to summarise the facts into a unified theory.

==List of decompositions==
* The [[Jordan–Chevalley decomposition]] of an element in algebraic group as a product of semisimple and unipotent elements
* The [[Bruhat decomposition]] <math>G=BWB</math> of a [[semisimple algebraic group]] into double [[coset]]s of a [[Borel subgroup]] can be regarded as a generalization of the principle of [[Gauss–Jordan elimination]], which generically writes a matrix as the product of an upper triangular matrix with a lower triangular matrix—but with exceptional cases. It is related to the Schubert cell decomposition of [[Grassmannian]]s: see [[Weyl group]] for more details.
*The [[Cartan decomposition]] writes a semisimple real Lie algebra as the sum of eigenspaces of a [[Cartan involution]].<ref>{{cite book |last=Kleiner |first=Israel |title=A History of Abstract Algebra |publisher=Birkhäuser |year=2007 |isbn=978-0817646844 |editor1-last=Kleiner |editor1-first=Israel |location=Boston, MA |doi=10.1007/978-0-8176-4685-1 |mr=2347309}}</ref>
* The [[Iwasawa decomposition]] <math>G=KAN</math> of a semisimple group <math>G</math> as the product of [[compact group|compact]], abelian, and [[nilpotent group|nilpotent]] subgroups generalises the way a square real matrix can be written as a product of an [[orthogonal matrix]] and an [[upper triangular matrix]] (a consequence of [[Gram–Schmidt orthogonalization]]).
*The [[Langlands decomposition]] <math>P=MAN</math> writes a parabolic subgroup <math>P</math> of a Lie group as the product of semisimple, abelian, and nilpotent subgroups.
* The [[Levi decomposition]] writes a finite dimensional Lie algebra as a [[semidirect product]] of a [[solvable Lie algebra|solvable]] ideal and a [[semisimple Lie algebra|semisimple]] subalgebra.
* The [[LU decomposition]] of a dense subset in the general linear group. It can be considered as a special case of the [[Bruhat decomposition]].
* The [[Birkhoff factorization|Birkhoff decomposition]], a special case of the [[Bruhat decomposition]] for affine groups.

==References==
{{reflist}}

[[Category:Lie groups]]
[[Category:factorization]]