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| In [[mathematics]], a '''compact''' ('''topological''', often understood) '''group''' is a [[topological group]] whose [[topology]] is [[compact space|compact]]. Compact groups are a natural generalisation of [[finite group]]s with the [[discrete topology]] and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to [[group action]]s and [[representation theory]].
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| In the following we will assume all groups are [[Hausdorff space]]s.
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| ==Compact Lie groups==
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| [[Lie group]]s form a very nice class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include
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| * the [[circle group]] '''T''' and the [[torus group]]s '''T'''<sup>''n''</sup>,
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| * the [[orthogonal group]]s O(''n''), the [[special orthogonal group]] SO(''n'') and its covering [[spin group]] Spin(''n''),
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| * the [[unitary group]] U(''n'') and the [[special unitary group]] SU(''n''),
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| * the [[symplectic group]] Sp(''n''),
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| * the compact forms of the [[exceptional Lie group]]s: [[G2 (mathematics)|G<sub>2</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], [[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], and [[E8 (mathematics)|E<sub>8</sub>]],
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| The [[classification theorem]] of compact Lie groups states that up to finite [[group extension|extensions]] and finite [[covering group|covers]] this exhausts the list of examples (which already includes some redundancies).
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| ===Classification===
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| Given any compact Lie group ''G'' one can take its [[identity component]] ''G''<sub>0</sub>, which is [[connected space|connected]]. The [[quotient group]] ''G''/''G''<sub>0</sub> is the group of components π<sub>0</sub>(''G'') which must be finite since ''G'' is compact. We therefore have a finite extension
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| :<math>1\to G_0 \to G \to \pi_0(G) \to 1.\,</math>
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| Now every compact, connected Lie group ''G''<sub>0</sub> has a finite covering
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| :<math>1\to A\to \tilde{G}_0\to G_0\to 1\,</math>
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| where <math>A\sub Z(\tilde{G}_0)</math> is a finite [[abelian group]] and <math>\tilde{G_0}</math> is a product of a torus and a compact, connected, simply-connected Lie group ''K'':
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| :<math>\tilde{G}_0 \cong \mathbb T^m \times K.</math>
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| Finally, every compact, connected, simply-connected Lie group ''K'' is a product of compact, connected, simply-connected [[simple Lie group]]s ''K''<sub>''i''</sub> each of which is isomorphic to exactly one of
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| *Sp(''n''), ''n'' ≥ 1
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| *SU(''n''), ''n'' ≥ 3
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| *Spin(''n''), ''n'' ≥ 7
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| [[G2 (mathematics)|G<sub>2</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], [[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], and [[E8 (mathematics)|E<sub>8</sub>]]
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| ==Further examples==
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| Amongst groups that are not Lie groups, and so do not carry the structure of a [[manifold]], examples are the additive group ''Z''<sub>''p''</sub> of [[p-adic integer]]s, and constructions from it. In fact any [[profinite group]] is a compact group. This means that [[Galois group]]s are compact groups, a basic fact for the theory of [[algebraic extension]]s in the case of infinite degree.
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| [[Pontryagin duality]] provides a large supply of examples of compact commutative groups. These are in duality with abelian [[discrete group]]s.
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| ==Haar measure==
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| Compact groups all carry a [[Haar measure]], which will be invariant by both left and right translation (the [[Haar measure|modulus function]] must be a continuous [[homomorphism]] to the positive multiplicative reals, and so 1). In other words these groups are [[unimodular group|unimodular]]. Haar measure is easily normalised to be a [[probability measure]], analogous to dθ/2π on the circle.
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| Such a Haar measure is in many cases easy to compute; for example for orthogonal groups it was known to [[Hurwitz]], and in the Lie group cases can always be given by an invariant [[differential form]]. In the profinite case there are many subgroups of [[finite index]], and Haar measure of a coset will be the reciprocal of the index. Therefore integrals are often computable quite directly, a fact applied constantly in [[number theory]].
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| ==Representation theory==
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| The representation theory of compact groups was founded by the [[Peter–Weyl theorem]]. [[Hermann Weyl]] went on to give the detailed [[character theory]] of the compact connected Lie groups, based on [[maximal torus]] theory. The resulting [[Weyl character formula]] was one of the influential results of twentieth century mathematics.
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| A combination of Weyl's work and [[Cartan's theorem]] gives a survey of the whole representation theory of compact groups ''G'' . That is, by the Peter–Weyl theorem the irreducible [[unitary representation]]s ρ of ''G'' are into a unitary group (of finite dimension) and the image will be a closed subgroup of the unitary group by compactness. Cartan's theorem states that Im(ρ) must itself be a Lie subgroup in the unitary group. If ''G'' is not itself a Lie group, there must be a kernel to ρ. Further one can form an [[inverse system]], for the kernel of ρ smaller and smaller, of finite-dimensional unitary representations, which identifies ''G'' as an [[inverse limit]] of compact Lie groups. Here the fact that in the limit a [[faithful representation]] of ''G'' is found is another consequence of the Peter–Weyl theorem,
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| The unknown part of the representation theory of compact groups is thereby, roughly speaking, thrown back onto the [[complex representations of finite groups]]. This theory is rather rich in detail, but is qualitatively well understood.
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| ==Duality==
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| The topic of recovering a compact group from its representation theory is the subject of the [[Tannaka–Krein duality]], now often recast in term of [[tannakian category]] theory.
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| ==From compact to non-compact groups==
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| The influence of the compact group theory on non-compact groups was formulated by Weyl in his [[unitarian trick]]. Inside a general [[semisimple Lie group]] there is a [[maximal compact subgroup]], and the representation theory of such groups, developed largely by [[Harish-Chandra]], uses intensively the [[restriction of a representation]] to such a subgroup, and also the model of Weyl's character theory.
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| ==See also==
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| *[[locally compact group]]
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| ==References==
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| *{{Citation
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| | last1 = Hofmann | first1 = Karl H. |authorlink1=Karl Heinrich Hofmann
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| | last2 = Morris | first2 = Sidney A. |authorlink2=Sidney Morris
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| | year = 1998
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| | title = The structure of compact groups
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| | publisher = de Gruyter
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| | location = Berlin
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| | isbn = 3-11-015268-1
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| }}
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| [[Category:Topological groups]]
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| [[Category:Lie groups]]
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| [[Category:Fourier analysis]]
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