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| In [[mathematics]], a '''maximal compact subgroup''' ''K'' of a [[topological group]] ''G'' is a [[subgroup]] ''K'' that is a [[compact space]], in the [[subspace topology]], and [[maximal]] amongst such subgroups.
| | == less than one meter tall == |
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| Maximal compact subgroups play an important role in the classification of Lie groups and especially semi-simple Lie groups. Maximal compact subgroups of Lie groups are ''not'' in general unique, but are unique up to [[Conjugacy_class#Conjugacy_of_subgroups_and_general_subsets|conjugation]] – they are [[essentially unique]].
| | London, such as the arrival of 布布鲁 of [http://www.nrcil.net/fancybox/lib/rakuten_LV_141.html ルイヴィトンの] being static.<br><br>'come.' Dylan whispered.<br><br>'Oh?' 罗峰立 that looked carefully, far across the sky [http://www.nrcil.net/fancybox/lib/rakuten_LV_42.html ルイヴィトン デザイナー] a golden streamer moment, then fell on the ground in front of Luo Feng, revealing the figure, which is a nearly six meters tall, strong hair a mess dwarves Han, who has a faint alcohol, his eyes restrained, only occasionally has an insane [http://www.nrcil.net/fancybox/lib/rakuten_LV_52.html ルイヴィトン 正規店] flashing.<br><br>Luo Feng mind a move.<br><br>Bubu Lu?<br>less than one meter tall<br>this six dwarves?<br><br>passing such an idea in mind though, you can still [http://www.nrcil.net/fancybox/lib/rakuten_LV_0.html ショルダーバッグ ルイヴィトン] respectful Luo Feng, teachers and security chiefs are not the same. Monkey-class status is also very high immortal gods, Luo Feng original Uncharted membership and the other that he [http://www.nrcil.net/fancybox/lib/rakuten_LV_29.html ルイヴィトン 財布 タイガ] could equal dialogue, but it has not the slightest [http://www.nrcil.net/fancybox/lib/rakuten_LV_22.html ルイヴィトン 長財布 中古] ability to suppress each other.<br><br>'You're Luo Feng?' dwarf brawny glanced Feng Luo.<br><br>'Luo Feng seen Bubu [http://www.nrcil.net/fancybox/lib/rakuten_LV_59.html ルイヴィトン 財布 通販] Lu teacher.' [http://www.nrcil.net/fancybox/lib/rakuten_LV_10.html ルイヴィトン ダミエ 新作] Luofeng Gong Jing salute. |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.ahyancheng.com/plus/feedback.php?aid=52 http://www.ahyancheng.com/plus/feedback.php?aid=52]</li> |
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| | <li>[http://www.cx930.net/home.php?mod=space&uid=138350 http://www.cx930.net/home.php?mod=space&uid=138350]</li> |
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| | <li>[http://www.kczj100.com/home.php?mod=space&uid=8377 http://www.kczj100.com/home.php?mod=space&uid=8377]</li> |
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| | </ul> |
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| ==Example== | | == I let him become a Warrior == |
| An example would be the subgroup O(2), the [[orthogonal group]], inside the [[general linear group]] GL(2, '''R'''). A related example is the [[circle group]] SO(2) inside [[SL2(R)|SL(2, '''R''')]]. Evidently SO(2) inside GL(2, '''R''') is compact and not maximal. The non-uniqueness of these examples can be seen as any [[inner product]] has an associated orthogonal group, and the essential uniqueness corresponds to the essential uniqueness of the inner product.
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| ==Definition==
| | A small number, no meaning.<br><br>'I have a high volume.' Luo [http://www.nrcil.net/fancybox/lib/rakuten_LV_58.html ルイヴィトン 靴] Feng has said.<br><br>group [http://www.nrcil.net/fancybox/lib/rakuten_LV_28.html ルイヴィトン 財布 定価] leaders a moment.<br><br>'big [http://www.nrcil.net/fancybox/lib/rakuten_LV_35.html ルイヴィトン 財布 一覧] quantities, can give many people use?' said a base city leaders from Africa, could not [http://www.nrcil.net/fancybox/lib/rakuten_LV_115.html ルイヴィトン アズール バッグ] help but [http://www.nrcil.net/fancybox/lib/rakuten_LV_16.html ルイヴィトン 限定バッグ] ask.<br><br>'world!'<br><br>low Luo Feng, 'I can [http://www.nrcil.net/fancybox/lib/rakuten_LV_79.html 財布 ルイヴィトン メンズ] supply all the world's advanced martial arts students, as long as the planet earth give birth to a 'higher martial arts students' taking purple gel, I let him become a Warrior!'<br><br>'all the world?'<br><br>'global supply?'<br><br>'Gee.'<br><br>'This, this ......'<br><br>many people know the truth, the more general the fewer number of treasures. Indeed, so large quantities of purple Luo Feng Lu Black Dragon, but spent 800 million dollars, an ordinary planet, as the [http://www.nrcil.net/fancybox/lib/rakuten_LV_98.html 大阪 ルイヴィトン] planet's richest man Prynne [http://www.nrcil.net/fancybox/lib/rakuten_LV_131.html ルイヴィトン 肩掛けバッグ] star so worth far less than this figure. Such large quantities of purple dew is not easy.<br><br>'higher total population and martial arts |
| A maximal compact subgroup is a maximal subgroup amongst proper compact subgroups – a ''maximal (compact subgroup)'' – rather than being (alternate possible reading) a [[maximal subgroup]] that happens to be compact; which would probably be called a ''compact (maximal subgroup)'', but in any case is not the intended meaning (and in fact maximal proper subgroups are not in general compact). | | 相关的主题文章： |
| | | <ul> |
| ==Existence and uniqueness==
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| The '''Cartan-Iwasawa-Malcev theorem''' asserts that every connected Lie group (and indeed every connected locally compact group) admits maximal compact subgroups and that they are all conjugate to one another. For a [[semisimple Lie group]] uniqueness is a consequence of the '''[[Cartan fixed point theorem]]''', which asserts that if a compact group acts by isometries on a complete simply connected [[negative curvature|negatively curved]] [[Riemannian manifold]] then it has a fixed point.
| | <li>[http://supraball.es/viewtopic.php?pid=211125#p211125 http://supraball.es/viewtopic.php?pid=211125#p211125]</li> |
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| Maximal compact subgroups of connected Lie groups are usually ''not'' unique, but they are unique up to conjugation, meaning that given two maximal compact subgroups ''K'' and ''L'', there is an element ''g'' ∈ ''G'' such that<ref>Note that this element ''g'' is not unique – any element in the same coset ''gK'' would do as well.</ref> ''gKg''<sup>-1</sup> = ''L'' – hence a maximal compact subgroup is [[essentially unique]], and people often speak of "the" maximal compact subgroup.
| | <li>[http://www.dp1234567.com/forum.php?mod=viewthread&tid=1427325 http://www.dp1234567.com/forum.php?mod=viewthread&tid=1427325]</li> |
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| For the example of the general linear group GL(''n'', '''R'''), this corresponds to the fact that ''any'' [[inner product]] on '''R'''<sup>''n''</sup> defines a (compact) orthogonal group (its isometry group) – and that it admits an orthonormal basis: the change of basis defines the conjugating element conjugating the isometry group to the classical orthogonal group O(''n'', '''R''').
| | <li>[http://202.206.64.233/darg/home.php?mod=space&uid=37213 http://202.206.64.233/darg/home.php?mod=space&uid=37213]</li> |
| ===Proofs===
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| For a real semisimple Lie group, Cartan's proof of the existence and uniqueness of a maximal compact subgroup can be found in {{harvtxt|Borel|1950}} and {{harvtxt|Helgason|1978}}. {{harvtxt|Cartier|1955}} and {{harvtxt|Hochschild|1965}} discuss the extension to connected Lie groups and connected locally compact groups.
| | </ul> |
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| For semisimple groups, existence is a consequence of the existence of a compact [[Complexification (Lie group)|real form]] of the noncompact semisimple Lie group and the corresponding [[Cartan decomposition]]. The proof of uniqueness relies on the fact that the corresponding [[Riemannian symmetric space]] ''G''/''K'' has [[negative curvature]] and
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| Cartan's fixed point theorem. {{harvtxt|Mostow|1955}} showed that the derivative of the exponential map at any point of ''G''/''K'' satisfies |d exp ''X''| ≥ |X|. This implies that ''G''/''K'' is a [[Hadamard space]], i.e. a [[complete metric space]] satisfying a weakened form of the parallelogram rule in a Euclidean space. Uniqueness can then be deduced from the [[Bruhat-Tits fixed point theorem]]. Indeed any bounded closed set in a Hadamard space is contained in a unique smallest closed ball, the center of which is called its [[circumcenter]]. In particular a compact group acting by isometries must fix the circumcenter of each of its orbits.
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| ===Proof of uniqueness for semisimple groups===
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| {{harvtxt|Mostow|1955}} also related the general problem for semisimple groups to the case of GL(''n'', '''R'''). The corresponding symmetric space is the space of positive symmetric matrices. A direct proof of uniqueness relying on elementary properties of this space is given in {{harvtxt|Hilgert|Neeb|2012}}.
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| Let <math>\mathfrak{g}</math> be a real semisimple Lie algebra with Cartan involution σ. Thus the fixed point subgroup of σ is the maximal compact subgroup ''K'' and there is an eigenspace decomposition
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| :<math>\displaystyle{\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p},}</math>
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| where <math>\mathfrak{k}</math>, the Lie algebra of ''K'', is the +1 eigenspace. The Cartan decomposition gives
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| :<math>\displaystyle{G=K\cdot \exp \mathfrak{p} = K\cdot P = P\cdot K.}</math>
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| If ''B'' is the [[Killing form]] on <math>\mathfrak{g}</math> given by ''B''(''X'',''Y'') = Tr (ad X)(ad Y), then
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| :<math>\displaystyle{(X,Y)_\sigma=-B(X,\sigma(Y))}</math> | |
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| is a real inner product on <math>\mathfrak{g}</math>. Under the adjoint representation, ''K'' is the subgroup of ''G'' that preserves this inner product.
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| If ''H'' is another compact subgroup of ''G'', then averaging the inner product over ''H'' with respect to the Haar measure gives an inner product invariant under ''H''. The operators Ad ''p'' with ''p'' in ''P'' are positive symmetric operators. This new inner produst can be written as
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| :<math>(S\cdot X,Y)_\sigma,</math>
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| where ''S'' is a positive symmetric operator on <math>\mathfrak{g}</math> such that
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| Ad(''h'')<sup>''t''</sup>''S'' Ad ''h'' = ''S'' for ''h'' in ''H'' (with the transposes computed with respect to the inner product). Moreover for ''x'' in ''G'',
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| :<math>\displaystyle{\mathrm{Ad}\, \sigma(x)=(\mathrm{Ad}\,(x)^{-1})^t.}</math>
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| So for ''h'' in ''H'',
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| :<math>\displaystyle{S\circ \mathrm{Ad}(\sigma(h))= \mathrm{Ad}(h)\circ S.}</math>
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| For ''X'' in <math>\mathfrak{p}</math> define
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| :<math>\displaystyle{f(e^X)=\mathrm{Tr}\, \mathrm{Ad}(e^X) S.}</math>
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| If ''e''<sub>''i''</sub> is an orthonormal basis of eigenvectors for ''S'' with ''Se''<sub>''i''</sub> = λ<sub>''i''</sub> ''e''<sub>''i''</sub>, then
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| :<math>\displaystyle{f(e^X)=\sum \lambda_i (\mathrm{Ad}(e^X)e_i,e_i)_\sigma \ge (\min \lambda_i)\cdot \mathrm{Tr}\,e^{\mathrm{ad}\,X},}</math>
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| so that ''f'' is strictly positive and tends to ∞ as |''X''| tends to ∞. In fact this norm is equivalent to the operator norm on the symmetric operators ad ''X'' and each non-zero eigenvalue occurs with its negative, since i ad ''X'' is a ''skew-adjoint operator'' on the compact real form <math>\mathfrak{k}\oplus i\mathfrak{p}</math>.
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| So ''f'' has a global minimum at ''Y'' say. This minimum is unique, because if ''Z'' were another then
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| :<math>\displaystyle{e^Z=e^{Y/2} e^X e^{Y/2},}</math>
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| where ''X'' in <math>\mathfrak{p}</math> is defined by the Cartan decomposition
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| :<math>\displaystyle{e^{Z/2}e^{-Y/2}=k\cdot e^{X/2}.}</math> | |
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| If ''f''<sub>''i''</sub> is an orthonormal basis of eigenvectors of ad ''X'' with corresponding real eigenvalues μ<sub>''i''</sub>, then
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| :<math>\displaystyle{g(t)= f(e^{Y/2} e^{tX} e^{Y/2})= \sum e^{\mu_i t} \|Ad(e^{Y/2})f_i\|^2_\sigma.}</math>
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| Since the right hand side is a positive combination of exponentials, the real-valued function ''g'' is [[strictly convex]] if ''X'' ≠ 0, so has a unique minimum. On the other hand it has local minima at ''t'' = 0 and ''t'' = 1, hence ''X'' = 0 and ''p'' = exp ''Y'' is the unique global minimum. By construction
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| ''f''(''x'') = ''f''(σ(''h'')''xh''<sup>−1</sup>) for ''h'' in ''H'', so that ''p'' = σ(''h'')''ph''<sup>−1</sup> for ''h'' in ''H''. Hence σ(''h'')= ''php''<sup>−1</sup>. Consequently, if ''g'' = exp ''Y''/2, ''gHg''<sup>−1</sup> is fixed by σ and therefore lies in ''K''.
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| ==Applications==
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| ===Representation theory===
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| Maximal compact subgroups play a basic role in the [[representation theory]] when ''G'' is not compact. In that case a maximal compact subgroup ''K'' is a [[compact Lie group]] (since a closed subgroup of a Lie group is a Lie group), for which the theory is easier.
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| The operations relating the representation theories of ''G'' and ''K'' are [[Restricted representation|restricting representations]] from ''G'' to ''K'', and [[induced representation|inducing representations]] from ''K'' to ''G,'', and these are quite well understood; their theory includes that of [[spherical function]]s.
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| ===Topology=== | |
| The [[algebraic topology]] of the Lie groups is also largely carried by a maximal compact subgroup ''K''. To be precise, a connected Lie group is a topological product (though not a group theoretic product!) of a maximal compact ''K'' and a Euclidean space – ''G'' = ''K'' × '''R'''<sup>''d''</sup> – thus in particular ''K'' is a [[deformation retract]] of ''G,'' and is [[homotopy equivalent]], and thus they have the same [[homotopy groups]]. Indeed, the inclusion <math>K \hookrightarrow G</math> and the deformation retraction <math>G \twoheadrightarrow K</math> are [[homotopy equivalence]]s.
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| For the general linear group, this decomposition is the [[QR decomposition]], and the deformation retraction is the [[Gram-Schmidt process]]. For a general semisimple Lie group, the decomposition is the [[Iwasawa decomposition]] of ''G'' as ''G'' = ''KAN'' in which ''K'' occurs in a product with a [[contractible]] subgroup ''AN''.
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| ==See also==
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| *[[Hyperspecial subgroup]]
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| *[[Complex Lie group]]
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| ==Notes==
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| {{reflist|1}}
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| ==References==
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| *{{citation|last=Borel|first= Armand|title=Sous-groupes compacts maximaux des groupes de Lie (Exposé No. 33)|series= Séminaire Bourbaki|volume=1|year=1950|url=http://www.numdam.org/numdam-bin/fitem?id=SB_1948-1951__1__271_0}}
| |
| *{{citation|last=Cartier|first=P.|title=
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| Structure topologique des groupes de Lie généraux (Exposé No. 22)|series= Séminaire "Sophus Lie"|volume= 1|year=1955|url=http://www.numdam.org/numdam-bin/fitem?id=SSL_1954-1955__1__A24_0}}
| |
| *{{citation|last=Dieudonné|first= J.|series =Treatise on analysis|title=Compact Lie groups and semisimple Lie groups, Chapter XXI|volume=5|publisher=Academic Press|year= 1977|isbn= 012215505X}}
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| * {{citation|first=Sigurdur|last=Helgason|title=Differential Geometry, Lie groups and Symmetric Spaces|publisher=Academic Press|year=1978|id=ISBN 978-0-12-338460-7}}
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| *{{citation|title=Structure and geometry of Lie groups|series=Springer monographs in mathematics|
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| last=Hilgert|first=Joachim|first2= Karl-Hermann|last2= Neeb|publisher=Springer|year= 2012|
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| id=ISBN 0387847944}}
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| *{{citation|last=Hochschild|first=G.|title=The structure of Lie groups|year=1965|publisher=Holden-Day}}
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| *{{citation|last=Mostow|first= G. D.|title= Some new decomposition theorems for semi-simple groups|series= Mem. Amer. Math. Soc. |year=1955|volume=14|pages= 31–54|url=http://archive.org/details/liealgebrasandli029541mbp}}
| |
| *{{citation|title=Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras|series=Encyclopaedia of Mathematical Sciences|volume= 41|last=Onishchik|first=A.L.|last2= Vinberg|first2= E.B.|year=1994|publisher=Springer|isbn=9783540546832}}
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| * {{citation|first=A.|last=Malcev|title=On the theory of Lie groups in the large|journal=Mat.Sbornik |volume= 16 |year=1945|pages=163-189}}
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| * {{citation|first=K.|last=Iwasawa|title=On some types of topological groups|journal= Ann. of Math.| volume=50|year=1949|pages= 507-558}}
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| [[Category:Topological groups]]
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| [[Category:Lie groups]]
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less than one meter tall
London, such as the arrival of 布布鲁 of ルイヴィトンの being static.
'come.' Dylan whispered.
'Oh?' 罗峰立 that looked carefully, far across the sky ルイヴィトン デザイナー a golden streamer moment, then fell on the ground in front of Luo Feng, revealing the figure, which is a nearly six meters tall, strong hair a mess dwarves Han, who has a faint alcohol, his eyes restrained, only occasionally has an insane ルイヴィトン 正規店 flashing.
Luo Feng mind a move.
Bubu Lu?
less than one meter tall
this six dwarves?
passing such an idea in mind though, you can still ショルダーバッグ ルイヴィトン respectful Luo Feng, teachers and security chiefs are not the same. Monkey-class status is also very high immortal gods, Luo Feng original Uncharted membership and the other that he ルイヴィトン 財布 タイガ could equal dialogue, but it has not the slightest ルイヴィトン 長財布 中古 ability to suppress each other.
'You're Luo Feng?' dwarf brawny glanced Feng Luo.
'Luo Feng seen Bubu ルイヴィトン 財布 通販 Lu teacher.' ルイヴィトン ダミエ 新作 Luofeng Gong Jing salute.
相关的主题文章：
I let him become a Warrior
A small number, no meaning.
'I have a high volume.' Luo ルイヴィトン 靴 Feng has said.
group ルイヴィトン 財布 定価 leaders a moment.
'big ルイヴィトン 財布 一覧 quantities, can give many people use?' said a base city leaders from Africa, could not ルイヴィトン アズール バッグ help but ルイヴィトン 限定バッグ ask.
'world!'
low Luo Feng, 'I can 財布 ルイヴィトン メンズ supply all the world's advanced martial arts students, as long as the planet earth give birth to a 'higher martial arts students' taking purple gel, I let him become a Warrior!'
'all the world?'
'global supply?'
'Gee.'
'This, this ......'
many people know the truth, the more general the fewer number of treasures. Indeed, so large quantities of purple Luo Feng Lu Black Dragon, but spent 800 million dollars, an ordinary planet, as the 大阪 ルイヴィトン planet's richest man Prynne ルイヴィトン 肩掛けバッグ star so worth far less than this figure. Such large quantities of purple dew is not easy.
'higher total population and martial arts
相关的主题文章：