Editing Maximal compact subgroup

{{Short description|Concept in topology}}
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 element|maximal]] amongst such subgroups.

Maximal compact subgroups play an important role in the classification of [[Lie group]]s 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]].

==Example==
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.

==Definition==
A maximal compact subgroup is a maximal subgroup amongst 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).

==Existence and uniqueness==
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|nonpositively curved]] [[Riemannian manifold]] then it has a fixed point.

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.

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''').

===Proofs===
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.

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 
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.

===Proof of uniqueness for semisimple groups===
{{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}}.

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

:<math>\displaystyle{\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p},}</math>

where <math>\mathfrak{k}</math>, the Lie algebra of ''K'', is the +1 eigenspace. The Cartan decomposition gives

:<math>\displaystyle{G=K\cdot \exp \mathfrak{p} = K\cdot P = P\cdot K.}</math>

If ''B'' is the [[Killing form]] on <math>\mathfrak{g}</math> given by ''B''(''X'',''Y'') = Tr (ad X)(ad Y), then

:<math>\displaystyle{(X,Y)_\sigma=-B(X,\sigma(Y))}</math>

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.

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

:<math>(S\cdot X,Y)_\sigma,</math>

where ''S'' is a positive symmetric operator on <math>\mathfrak{g}</math> such that 
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'',

:<math>\displaystyle{\mathrm{Ad}\, \sigma(x)=(\mathrm{Ad}\,(x)^{-1})^t.}</math>

So for ''h'' in ''H'',

:<math>\displaystyle{S\circ \mathrm{Ad}(\sigma(h))= \mathrm{Ad}(h)\circ S.}</math>

For ''X'' in <math>\mathfrak{p}</math> define

:<math>\displaystyle{f(e^X)=\mathrm{Tr}\, \mathrm{Ad}(e^X) S.}</math>

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

:<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>

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>.

So ''f'' has a global minimum at ''Y'' say. This minimum is unique, because if ''Z'' were another then

:<math>\displaystyle{e^Z=e^{Y/2} e^X e^{Y/2},}</math>

where ''X'' in <math>\mathfrak{p}</math> is defined by the Cartan decomposition

:<math>\displaystyle{e^{Z/2}e^{-Y/2}=k\cdot e^{X/2}.}</math>

If ''f''<sub>''i''</sub> is an orthonormal basis of eigenvectors of ad ''X'' with corresponding real eigenvalues μ<sub>''i''</sub>, then

:<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>

Since the right hand side is a positive combination of exponentials, the real-valued function ''g'' is [[Strictly convex function|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
''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''.

==Applications==

===Representation theory===
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.

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 [[Zonal spherical function|spherical function]]s.

===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.

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''.

==See also==
*[[Hyperspecial subgroup]]
*[[Complex Lie group]]

==Notes==
{{reflist|1}}

==References==
*{{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/item/SB_1948-1951__1__271_0/}}
*{{citation|last=Cartier|first=P.|title= 
Structure topologique des groupes de Lie généraux (Exposé No. 22)|series= Séminaire "Sophus Lie"|volume= 1|year=1955|pages=1–20 |url=http://www.numdam.org/item/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}}
* {{citation|first=Sigurdur|last=Helgason|title=Differential Geometry, Lie groups and Symmetric Spaces|publisher=Academic Press|year=1978|isbn=978-0-12-338460-7}}
*{{citation|title=Structure and geometry of Lie groups|series=Springer monographs in mathematics|
last1=Hilgert|first1=Joachim|first2= Karl-Hermann|last2= Neeb|publisher=Springer|year= 2012|
isbn=978-0387847948}}
*{{citation|last=Hochschild|first=G.|title=The structure of Lie groups|year=1965|publisher=Holden-Day}}
*{{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|publisher= American Mathematical Society|url=https://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|last1=Onishchik|first1=A.L.|last2= Vinberg|first2= E.B.|year=1994|publisher=Springer|isbn=9783540546832}}
* {{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}}
* {{citation|first=K.|last=Iwasawa|title=On some types of topological groups|journal= Ann. of Math.| volume=50|year=1949|issue=3 |pages= 507–558|doi=10.2307/1969548|jstor=1969548 }}

[[Category:Topological groups]]
[[Category:Lie groups]]