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A [[Topological group|topological group]] that is compact as a topological space. For example, every finite group (in the discrete topology) is a compact group. An algebraic group, even though it is a compact topological space (with respect to the Zariski topology), is not a topological group with respect to this topology and therefore is not a compact group.
 
A [[Topological group|topological group]] that is compact as a topological space. For example, every finite group (in the discrete topology) is a compact group. An algebraic group, even though it is a compact topological space (with respect to the Zariski topology), is not a topological group with respect to this topology and therefore is not a compact group.
  
 
The following groups are two important classes of compact groups.
 
The following groups are two important classes of compact groups.
  
1) Locally connected compact groups. Examples of such compact groups are the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c0234701.png" /> of all unitary complex matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c0234702.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c0234703.png" /> of all orthogonal real matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c0234704.png" /> (with the topology induced by the topology determined by the ordinary norm of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c0234705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c0234706.png" />, respectively) and, more generally, any compact real Lie group.
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1) Locally connected compact groups. Examples of such compact groups are the group $  U ( n ,\  \mathbf C ) $  of all unitary complex matrices of order $  n $ , the group $  \textrm{ O }( n ,\  \mathbf R ) $  of all orthogonal real matrices of order $  n $  (with the topology induced by the topology determined by the ordinary norm of the fields $  \mathbf C $  and $  \mathbf R $ , respectively) and, more generally, any compact real Lie group.
  
2) Totally-disconnected compact groups. Of this type is the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c0234707.png" /> of invertible matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c0234708.png" /> with coefficients in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c0234709.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347010.png" />-adic integers (with the topology induced by that determined by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347011.png" />-adic norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347012.png" />; see [[Totally-disconnected space|Totally-disconnected space]]).
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2) Totally-disconnected compact groups. Of this type is the group $  \mathop{\rm GL}\nolimits ( n ,\  \mathbf Z _{p} ) $  of invertible matrices of order $  n $  with coefficients in the ring $  \mathbf Z _{p} $  of $  p $ -adic integers (with the topology induced by that determined by the $  p $ -adic norm of $  \mathbf Z _{p} $ ; see [[Totally-disconnected space|Totally-disconnected space]]).
  
Any totally-disconnected compact group is a [[Profinite group|profinite group]], and conversely, every profinite group is a totally-disconnected compact group. The totally-disconnected compact Hausdorff groups can be characterized as the compact groups of topological dimension zero. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347013.png" /> is locally connected and finite-dimensional, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347014.png" /> is a real Lie group [[#References|[1]]]. The structure of a compact group of general type is to a certain degree determined by the structure of these two types of compact groups. In an arbitrary finite-dimensional compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347015.png" /> there is zero-dimensional subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347016.png" /> (lying in the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347017.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347018.png" /> is a real Lie group and, furthermore, some neighbourhood of the identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347019.png" /> is the direct product of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347020.png" /> and a real local Lie group (cf. [[Lie group, local|Lie group, local]]). Every connected finite-dimensional compact group has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347022.png" /> is a simply-connected compact semi-simple real Lie group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347023.png" /> is a finite-dimensional connected commutative compact group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347024.png" /> is a finite central normal subgroup for which only the identity lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347025.png" />. The study of the structure of connected compact real Lie groups has led to a complete classification of them (see [[Lie group, compact|Lie group, compact]]); the structure of commutative compact groups is elucidated in the theory of Pontryagin duality. Any compact group (not necessarily finite-dimensional) is the projective limit of compact real Lie groups [[#References|[2]]]. The topological structure of the above two types of compact groups is as follows: Every locally connected finite-dimensional compact group is a topological manifold, while every infinite zero-dimensional compact group with a countable base is homeomorphic to the perfect Cantor set.
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Any totally-disconnected compact group is a [[Profinite group|profinite group]], and conversely, every profinite group is a totally-disconnected compact group. The totally-disconnected compact Hausdorff groups can be characterized as the compact groups of topological dimension zero. If $  G $  is locally connected and finite-dimensional, then $  G $  is a real Lie group [[#References|[1]]]. The structure of a compact group of general type is to a certain degree determined by the structure of these two types of compact groups. In an arbitrary finite-dimensional compact group $  G $  there is zero-dimensional subgroup $  N $  (lying in the centre of $  G $ ) such that $  G / N $  is a real Lie group and, furthermore, some neighbourhood of the identity in $  G $  is the direct product of the group $  N $  and a real local Lie group (cf. [[Lie group, local|Lie group, local]]). Every connected finite-dimensional compact group has the form $  ( P \times C ) / Z $ , where $  P $  is a simply-connected compact semi-simple real Lie group, $  C $  is a finite-dimensional connected commutative compact group and $  Z $  is a finite central normal subgroup for which only the identity lies in $  C $ . The study of the structure of connected compact real Lie groups has led to a complete classification of them (see [[Lie group, compact|Lie group, compact]]); the structure of commutative compact groups is elucidated in the theory of Pontryagin duality. Any compact group (not necessarily finite-dimensional) is the projective limit of compact real Lie groups [[#References|[2]]]. The topological structure of the above two types of compact groups is as follows: Every locally connected finite-dimensional compact group is a topological manifold, while every infinite zero-dimensional compact group with a countable base is homeomorphic to the perfect Cantor set.
  
The study of structure of compact groups is based on the fact that every compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347026.png" /> has a sufficient system of finite-dimensional linear representations, that is, for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347027.png" /> there exists a continuous finite-dimensional linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347028.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347029.png" />. This fact is one of the important results of the well-developed general theory of linear representations of compact groups. This theory makes essential use of the fact that every compact group has a two-sided invariant measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347030.png" /> (a [[Haar measure|Haar measure]]), which enables one to define invariant integration on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347031.png" />. The most important facts of this theory are as follows. Every continuous representation of a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347032.png" /> in a pre-Hilbert space is equivalent to a unitary representation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347033.png" /> be the Hilbert space of square-integrable complex-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347034.png" /> with respect to the invariant measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347035.png" />. The action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347036.png" /> on the functions by left and right translations determines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347037.png" /> the structure of a left and a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347038.png" />-module. The corresponding representations are respectively called the left and right regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347039.png" />; they are unitary and unitarily equivalent. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347040.png" /> be the family of all possible pairwise inequivalent finite-dimensional irreducible unitary representations of the compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347041.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347043.png" />, be the set of matrix elements of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347044.png" /> in some orthonormal basis. Then the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347045.png" /> lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347046.png" /> and form in it a complete orthogonal system, the norm of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347047.png" /> being <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347048.png" />. Any continuous complex-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347049.png" /> can, to any desired degree of accuracy, be uniformly approximated by finite linear combinations of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347050.png" /> (the Peter–Weyl theorem). The characters of the irreducible unitary finite-dimensional representations are pairwise orthogonal and have norm 1. Continuous finite-dimensional unitary representations are equivalent if and only if their characters are equal. A continuous finite-dimensional unitary representation is irreducible if and only if the norm of its character (which lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347051.png" />) is equal to 1. Every irreducible continuous unitary representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347052.png" /> in a Hilbert space is finite-dimensional. Every continuous unitary representation of the group in a Hilbert space is an orthogonal direct sum of unitary representations that are multiples of finite-dimensional irreducible representations. In particular, the multiplicity of the imbedding of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347053.png" /> in the right regular representation is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347054.png" />; furthermore, the sum of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347055.png" />-submodules in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347056.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347057.png" /> that are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347058.png" /> is precisely the linear span of all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023470/c02347059.png" />.
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The study of structure of compact groups is based on the fact that every compact group $  G $  has a sufficient system of finite-dimensional linear representations, that is, for any element $  g \in G $  there exists a continuous finite-dimensional linear representation $  \rho $  for which $  g \notin  \mathop{\rm Ker}\nolimits \  \rho $ . This fact is one of the important results of the well-developed general theory of linear representations of compact groups. This theory makes essential use of the fact that every compact group has a two-sided invariant measure $  \mu (g) $  (a [[Haar measure|Haar measure]]), which enables one to define invariant integration on $  G $ . The most important facts of this theory are as follows. Every continuous representation of a compact group $  G $  in a pre-Hilbert space is equivalent to a unitary representation. Let $  L _{2} (G) $  be the Hilbert space of square-integrable complex-valued functions on $  G $  with respect to the invariant measure $  \mu (g) $ . The action of the group $  G $  on the functions by left and right translations determines on $  L _{2} (G) $  the structure of a left and a right $  G $ -module. The corresponding representations are respectively called the left and right regular representation of $  G $ ; they are unitary and unitarily equivalent. Let $  \{ {R ^ \alpha } : {\alpha \in I} \} $  be the family of all possible pairwise inequivalent finite-dimensional irreducible unitary representations of the compact group $  G $  and let $  m _{ij} ^ \alpha  (g) $ ,  $  i ,\  j = 1 \dots n _ \alpha  = \mathop{\rm dim}\nolimits \  R ^ \alpha  $ , be the set of matrix elements of the representation $  R ^ \alpha  $  in some orthonormal basis. Then the functions $  m _{ij} ^ \alpha  (g) $  lie in $  L _{2} (G) $  and form in it a complete orthogonal system, the norm of the function $  m _{ij} ^ \alpha  (g) $  being $  n _ \alpha  ^{-1}/2 $ . Any continuous complex-valued function on $  G $  can, to any desired degree of accuracy, be uniformly approximated by finite linear combinations of the functions $  m _{ij} ^ \alpha  (g) $  (the Peter–Weyl theorem). The characters of the irreducible unitary finite-dimensional representations are pairwise orthogonal and have norm 1. Continuous finite-dimensional unitary representations are equivalent if and only if their characters are equal. A continuous finite-dimensional unitary representation is irreducible if and only if the norm of its character (which lies in $  L _{2} (G) $ ) is equal to 1. Every irreducible continuous unitary representation of the group $  G $  in a Hilbert space is finite-dimensional. Every continuous unitary representation of the group in a Hilbert space is an orthogonal direct sum of unitary representations that are multiples of finite-dimensional irreducible representations. In particular, the multiplicity of the imbedding of the representation $  R ^ \alpha  $  in the right regular representation is equal to $  n _ \alpha  = \mathop{\rm dim}\nolimits \  R ^ \alpha  $ ; furthermore, the sum of all $  G $ -submodules in the $  G $ -module $  L _{2} (G) $  that are isomorphic to $  R ^ \alpha  $  is precisely the linear span of all the $  m _{ij} ^ \alpha  (g) $ .
  
 
====References====
 
====References====

Revision as of 18:04, 12 December 2019

A topological group that is compact as a topological space. For example, every finite group (in the discrete topology) is a compact group. An algebraic group, even though it is a compact topological space (with respect to the Zariski topology), is not a topological group with respect to this topology and therefore is not a compact group.

The following groups are two important classes of compact groups.

1) Locally connected compact groups. Examples of such compact groups are the group $ U ( n ,\ \mathbf C ) $ of all unitary complex matrices of order $ n $ , the group $ \textrm{ O }( n ,\ \mathbf R ) $ of all orthogonal real matrices of order $ n $ (with the topology induced by the topology determined by the ordinary norm of the fields $ \mathbf C $ and $ \mathbf R $ , respectively) and, more generally, any compact real Lie group.

2) Totally-disconnected compact groups. Of this type is the group $ \mathop{\rm GL}\nolimits ( n ,\ \mathbf Z _{p} ) $ of invertible matrices of order $ n $ with coefficients in the ring $ \mathbf Z _{p} $ of $ p $ -adic integers (with the topology induced by that determined by the $ p $ -adic norm of $ \mathbf Z _{p} $ ; see Totally-disconnected space).

Any totally-disconnected compact group is a profinite group, and conversely, every profinite group is a totally-disconnected compact group. The totally-disconnected compact Hausdorff groups can be characterized as the compact groups of topological dimension zero. If $ G $ is locally connected and finite-dimensional, then $ G $ is a real Lie group [1]. The structure of a compact group of general type is to a certain degree determined by the structure of these two types of compact groups. In an arbitrary finite-dimensional compact group $ G $ there is zero-dimensional subgroup $ N $ (lying in the centre of $ G $ ) such that $ G / N $ is a real Lie group and, furthermore, some neighbourhood of the identity in $ G $ is the direct product of the group $ N $ and a real local Lie group (cf. Lie group, local). Every connected finite-dimensional compact group has the form $ ( P \times C ) / Z $ , where $ P $ is a simply-connected compact semi-simple real Lie group, $ C $ is a finite-dimensional connected commutative compact group and $ Z $ is a finite central normal subgroup for which only the identity lies in $ C $ . The study of the structure of connected compact real Lie groups has led to a complete classification of them (see Lie group, compact); the structure of commutative compact groups is elucidated in the theory of Pontryagin duality. Any compact group (not necessarily finite-dimensional) is the projective limit of compact real Lie groups [2]. The topological structure of the above two types of compact groups is as follows: Every locally connected finite-dimensional compact group is a topological manifold, while every infinite zero-dimensional compact group with a countable base is homeomorphic to the perfect Cantor set.

The study of structure of compact groups is based on the fact that every compact group $ G $ has a sufficient system of finite-dimensional linear representations, that is, for any element $ g \in G $ there exists a continuous finite-dimensional linear representation $ \rho $ for which $ g \notin \mathop{\rm Ker}\nolimits \ \rho $ . This fact is one of the important results of the well-developed general theory of linear representations of compact groups. This theory makes essential use of the fact that every compact group has a two-sided invariant measure $ \mu (g) $ (a Haar measure), which enables one to define invariant integration on $ G $ . The most important facts of this theory are as follows. Every continuous representation of a compact group $ G $ in a pre-Hilbert space is equivalent to a unitary representation. Let $ L _{2} (G) $ be the Hilbert space of square-integrable complex-valued functions on $ G $ with respect to the invariant measure $ \mu (g) $ . The action of the group $ G $ on the functions by left and right translations determines on $ L _{2} (G) $ the structure of a left and a right $ G $ -module. The corresponding representations are respectively called the left and right regular representation of $ G $ ; they are unitary and unitarily equivalent. Let $ \{ {R ^ \alpha } : {\alpha \in I} \} $ be the family of all possible pairwise inequivalent finite-dimensional irreducible unitary representations of the compact group $ G $ and let $ m _{ij} ^ \alpha (g) $ , $ i ,\ j = 1 \dots n _ \alpha = \mathop{\rm dim}\nolimits \ R ^ \alpha $ , be the set of matrix elements of the representation $ R ^ \alpha $ in some orthonormal basis. Then the functions $ m _{ij} ^ \alpha (g) $ lie in $ L _{2} (G) $ and form in it a complete orthogonal system, the norm of the function $ m _{ij} ^ \alpha (g) $ being $ n _ \alpha ^{-1}/2 $ . Any continuous complex-valued function on $ G $ can, to any desired degree of accuracy, be uniformly approximated by finite linear combinations of the functions $ m _{ij} ^ \alpha (g) $ (the Peter–Weyl theorem). The characters of the irreducible unitary finite-dimensional representations are pairwise orthogonal and have norm 1. Continuous finite-dimensional unitary representations are equivalent if and only if their characters are equal. A continuous finite-dimensional unitary representation is irreducible if and only if the norm of its character (which lies in $ L _{2} (G) $ ) is equal to 1. Every irreducible continuous unitary representation of the group $ G $ in a Hilbert space is finite-dimensional. Every continuous unitary representation of the group in a Hilbert space is an orthogonal direct sum of unitary representations that are multiples of finite-dimensional irreducible representations. In particular, the multiplicity of the imbedding of the representation $ R ^ \alpha $ in the right regular representation is equal to $ n _ \alpha = \mathop{\rm dim}\nolimits \ R ^ \alpha $ ; furthermore, the sum of all $ G $ -submodules in the $ G $ -module $ L _{2} (G) $ that are isomorphic to $ R ^ \alpha $ is precisely the linear span of all the $ m _{ij} ^ \alpha (g) $ .

References

[1] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[2] A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940) MR0005741
[3] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[4] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013


Comments

References

[a1] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1–2 , Springer (1970) MR0262773 Zbl 0213.40103
[a2] D. Montgomery, L. Zippin, "Topological transformation groups" , Interscience (1955) MR0073104 Zbl 0068.01904
[a3] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
How to Cite This Entry:
Compact group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Compact_group&oldid=44230
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article