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In [[mathematics]], the '''Iwasawa decomposition''' KAN of a [[semisimple Lie group]] 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]]). It is named after [[Kenkichi Iwasawa]], the [[Japan]]ese [[mathematician]] who developed this method.

==Definition==
*''G'' is a connected semisimple real [[Lie group]]. 
*<math> \mathfrak{g}_0 </math> is the [[Lie algebra]] of ''G''
*<math> \mathfrak{g} </math>  is the [[complexification]] of <math> \mathfrak{g}_0 </math>.
*θ is a [[Cartan involution]] of <math> \mathfrak{g}_0 </math>
*<math> \mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{p}_0 </math> is the corresponding [[Cartan decomposition]] 
*<math> \mathfrak{a}_0 </math> is a maximal abelian subalgebra of <math> \mathfrak{p}_0 </math>
*Σ is the set of restricted roots of <math> \mathfrak{a}_0 </math>, corresponding to eigenvalues of <math> \mathfrak{a}_0 </math> acting on <math> \mathfrak{g}_0 </math>.
*Σ<sup>+</sup> is a choice of positive roots of Σ 
*<math> \mathfrak{n}_0 </math> is a nilpotent Lie algebra given as the  sum of the root spaces of Σ<sup>+</sup>
*''K'', ''A'', ''N'', are the Lie subgroups of ''G'' generated by <math> \mathfrak{k}_0, \mathfrak{a}_0 </math> and <math> \mathfrak{n}_0 </math>.

Then the '''Iwasawa decomposition''' of <math> \mathfrak{g}_0 </math> is
:<math>\mathfrak{g}_0 = \mathfrak{k}_0 + \mathfrak{a}_0 + \mathfrak{n}_0</math>
and the Iwasawa decomposition of ''G'' is 
:<math>G=KAN</math>

The [[dimension]] of ''A'' (or equivalently of <math> \mathfrak{a}_0 </math>) is called the '''real rank''' of ''G''.

Iwasawa decompositions also hold for some disconnected semisimple groups ''G'', where ''K'' becomes a (disconnected) [[maximal compact subgroup]] provided the center of ''G'' is finite.

The restricted root space decomposition is
:<math> \mathfrak{g}_0 = \mathfrak{m}_0\oplus\mathfrak{a}_0\oplus_{\lambda\in\Sigma}\mathfrak{g}_{\lambda} </math>
where  <math>\mathfrak{m}_0</math> is the centralizer of <math>\mathfrak{a}_0</math> in <math>\mathfrak{k}_0</math> and <math>\mathfrak{g}_{\lambda} = \{X\in\mathfrak{g}_0: [H,X]=\lambda(H)X\;\;\forall H\in\mathfrak{a}_0 \}</math> is the root space. The number 
<math>m_{\lambda}= \text{dim}\,\mathfrak{g}_{\lambda}</math> is called the multiplicity of <math>\lambda</math>.

==Examples==
If ''G''=''GL<sub>n''</sub>('''R'''), then we can take ''K'' to be the orthogonal matrices, ''A'' to be the positive diagonal matrices, and ''N'' to be the [[unipotent group]] consisting of upper triangular matrices with 1s on the diagonal.

==Non-archimedian Iwasawa decomposition ==
There is an analogon to the above Iwasawa decomposition for a [[non-archimedean field]] ''F'': In this case, the group <math>GL_n(F)</math> can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup <math>GL_n(O_F)</math>, where <math>O_F</math> is the [[ring of integers]] of ''F''.
<ref>Bump, Automorphic Forms and Representations, Prop. 4.5.2</ref>

==See also==
*[[Lie group decompositions]]



==References==
*{{springer|id=I/i053060|first1=A.S. |last1=Fedenko|first2=A.I.|last2= Shtern}}
*[[A. W. Knapp]], ''Structure theory of semisimple Lie groups'', in ISBN 0-8218-0609-2: Representation Theory and Automorphic Forms: Instructional Conference, International Centre for Mathematical Sciences, March 1996, Edinburgh, Scotland (Proceedings of Symposia in Pure Mathematics) by T. N. Bailey (Editor), Anthony W. Knapp (Editor)

*[[Kenkichi Iwasawa|Iwasawa, Kenkichi]]: On some types of topological groups. Annals of Mathematics (2) 50, (1949), 507&ndash;558.
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[[Category:Lie groups]]