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Iwasawa decomposition
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{{Short description|Mathematical process dealing with Lie groups}} In [[mathematics]], the '''Iwasawa decomposition''' (aka '''KAN''' from its expression) 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]] ([[QR decomposition]], a consequence of [[Gram–Schmidt process|Gram–Schmidt orthogonalization]]). It is named after [[Kenkichi Iwasawa]], the [[Japan]]ese [[mathematician]] who developed this method.<ref>{{cite journal |authorlink=Kenkichi Iwasawa |last=Iwasawa |first=Kenkichi |title=On Some Types of Topological Groups |journal=[[Annals of Mathematics]] |series=<!-- Second series --> |volume=50 |year=1949 |issue=3 |pages=507–558 |jstor=1969548 |doi=10.2307/1969548}}</ref> ==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 \oplus \mathfrak{a}_0 \oplus \mathfrak{n}_0</math> and the Iwasawa decomposition of ''G'' is :<math>G=KAN</math> meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold <math> K \times A \times N </math> to the Lie group <math> G </math>, sending <math> (k,a,n) \mapsto kan </math>. The [[dimension]] of ''A'' (or equivalently of <math> \mathfrak{a}_0 </math>) is equal to the [[Algebraic torus#Flat subspaces and rank of symmetric spaces|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''=''SL<sub>n</sub>''('''R'''), then we can take ''K'' to be the orthogonal matrices, ''A'' to be the positive diagonal matrices with determinant 1, and ''N'' to be the [[unipotent group]] consisting of upper triangular matrices with 1s on the diagonal. For the case of ''n'' = 2, the Iwasawa decomposition of ''G'' = ''SL''(2, '''R''') is in terms of :<math> \mathbf{K} = \left\{ \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \in SL(2,\mathbb{R}) \ | \ \theta\in\mathbf{R} \right\} \cong SO(2) , </math> :<math> \mathbf{A} = \left\{ \begin{pmatrix} r & 0 \\ 0 & r^{-1} \end{pmatrix} \in SL(2,\mathbb{R}) \ | \ r > 0 \right\}, </math> :<math> \mathbf{N} = \left\{ \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix} \in SL(2,\mathbb{R}) \ | \ x\in\mathbf{R} \right\}. </math> For the [[symplectic group]] ''G'' = ''Sp''(2''n'', '''R'''), a possible Iwasawa decomposition is in terms of :<math> \mathbf{K} = Sp(2n,\mathbb{R})\cap SO(2n) = \left\{ \begin{pmatrix} A & B \\ -B & A \end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ A+iB \in U(n) \right\} \cong U(n) , </math> :<math> \mathbf{A} = \left\{ \begin{pmatrix} D & 0 \\ 0 & D^{-1} \end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ D \text{ positive, diagonal} \right\}, </math> :<math> \mathbf{N} = \left\{ \begin{pmatrix} N & M \\ 0 & N^{-T} \end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ N \text{ upper triangular with diagonal elements = 1},\ NM^T = MN^T \right\}. </math> ==Non-Archimedean Iwasawa decomposition== There is an analog to the above Iwasawa decomposition for a [[non-Archimedean field]] <math>F</math>: 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 <math>F</math>.<ref>{{citation|author=Bump|first=Daniel|title=Automorphic forms and representations|publisher=Cambridge University Press|location=Cambridge|year=1997|isbn=0-521-55098-X|doi=10.1017/CBO9780511609572}}, Prop. 4.5.2</ref> ==See also== * [[Lie group decompositions]] * [[Root system of a semi-simple Lie algebra]] ==References== {{Reflist}} *{{springer|title=Iwasawa decomposition|id=Iwasawa_decomposition&oldid=21877|first1=A.S. |last1=Fedenko|first2=A.I.|last2= Shtern}} *{{Cite book|title=Lie groups beyond an introduction|authorlink=A. W. Knapp|last=Knapp|first=A. W.|isbn=9780817642594|year=2002|publisher=Springer |edition=2nd}} [[Category:Lie groups]]
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