Differential of the first kind: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>LokiClock
 
en>JP.Martin-Flatin
The spelling "hyperelliptic" is far more frequent than the spelling "hyper-elliptic"
Line 1: Line 1:
Myrtle Benny is how I'm called and I really feel comfy when individuals use the full name. To gather coins is what her family members and her appreciate. For years I've been operating as a payroll clerk. Years ago we moved to North Dakota and I adore every working day residing here.<br><br>Feel free to visit my blog :: [http://nuvem.tk/altergalactica/MarissaepDonnt nuvem.tk]
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.
{{Reflist}}
 
 
[[Category:Lie groups]]

Revision as of 12:21, 13 September 2013

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 Japanese mathematician who developed this method.

Definition

Then the Iwasawa decomposition of is

and the Iwasawa decomposition of G is

The dimension of A (or equivalently of ) 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

where is the centralizer of in and is the root space. The number is called the multiplicity of .

Examples

If G=GLn(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 can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup , where is the ring of integers of F. [1]

See also


References

  • Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

    my web-site http://himerka.com/
  • 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)
  • Iwasawa, Kenkichi: On some types of topological groups. Annals of Mathematics (2) 50, (1949), 507–558.

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

  1. Bump, Automorphic Forms and Representations, Prop. 4.5.2