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| In [[mathematics]], the '''Peter–Weyl theorem''' is a basic result in the theory of [[harmonic analysis]], applying to [[topological group]]s that are [[Compact group|compact]], but are not necessarily [[Abelian group|abelian]]. It was initially proved by [[Hermann Weyl]], with his student [[Fritz Peter]], in the setting of a compact [[topological group]] ''G'' {{harv|Peter|Weyl|1927}}. The theorem is a collection of results generalizing the significant facts about the decomposition of the [[regular representation]] of any [[finite group]], as discovered by [[F. G. Frobenius]] and [[Issai Schur]].
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| The theorem has three parts. The first part states that the matrix coefficients of [[irreducible representation]]s of ''G'' are dense in the space ''C''(''G'') of continuous [[complex-valued function]]s on ''G'', and thus also in the space ''L''<sup>2</sup>(''G'') of [[square-integrable function]]s. The second part asserts the complete reducibility of [[unitary representation]]s of ''G''. The third part then asserts that the regular representation of ''G'' on ''L''<sup>2</sup>(''G'') decomposes as the direct sum of all irreducible unitary representations. Moreover, the matrix coefficients of the irreducible unitary representations form an [[orthonormal basis]] of ''L''<sup>2</sup>(''G'').
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| ==Matrix coefficients== | |
| A '''[[matrix coefficient]]''' of the group ''G'' is a complex-valued function φ on ''G'' given as the composition
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| :<math>\varphi = L\circ \pi</math>
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| where π : ''G'' → GL(''V'') is a finite-dimensional ([[continuous function|continuous]]) [[group representation]] of ''G'', and ''L'' is a [[linear functional]] on the vector space of [[endomorphism]]s of ''V'' (e.g. trace), which contains GL(''V'') as an open subset. Matrix coefficients are continuous, since representations are by definition continuous, and linear functionals on finite-dimensional spaces are also continuous.
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| The first part of the Peter–Weyl theorem asserts ({{harvnb|Bump|2004|loc=§4.1}}; {{harvnb|Knapp|1986|loc=Theorem 1.12}}):
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| <blockquote>'''Peter-Weyl Theorem (Part I).''' The set of matrix coefficients of ''G'' is [[dense set|dense]] in the space of [[continuous functions on a compact Hausdorff space|continuous complex functions]] C(''G'') on ''G'', equipped with the [[uniform norm]].</blockquote>
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| This first result resembles the [[Stone-Weierstrass theorem]] in that it indicates the density of a set of functions in the space of all continuous functions, subject only to an ''algebraic'' characterization. In fact, if ''G'' is a [[matrix group]], then the result follows easily from the Stone-Weierstrass theorem {{harv|Knapp|1986|p=17}}. Conversely, it is a consequence of the theorem that any compact [[Lie group]] is isomorphic to a matrix group {{harv|Knapp|1986|loc=Theorem 1.15}}.
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| A corollary of this result is that the matrix coefficients of ''G'' are dense in ''L''<sup>2</sup>(''G'').
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| ==Decomposition of a unitary representation==
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| The second part of the theorem gives the existence of a decomposition of a [[unitary representation]] of ''G'' into finite-dimensional representations. Now, intuitively groups were conceived as rotations on geometric objects, so it is only natural to study representations which essentially arise from continuous '''[[group action|actions]]''' on Hilbert spaces. (For those who were first introduced to dual groups consisting of characters which are the continuous homomorphisms into the [[circle group]], this approach is similar except that the circle group is (ultimately) generalised to the group of unitary operators on a given Hilbert space.)
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| Let ''G'' be a topological group and ''H'' a complex Hilbert space.
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| A continuous action ∗ : ''G'' × ''H'' → ''H'', gives rise to a continuous map ρ<sub>∗</sub> : ''G'' → ''H''<sup>''H''</sup> (functions from ''H'' to ''H'' with the [[strong topology]]) defined by: ρ<sub>∗</sub>(''v'') = ''∗(g,v)''. This map is clearly an homomorphism from ''G'' into GL(''H''), the homeomorphic automorphisms of ''H''. Conversely, given such a map, we can uniquely recover the action in the obvious way.
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| Thus we define the '''representations of ''G'' on an Hilbert space ''H''''' to be those [[group homomorphisms]], ρ, which arise from continuous actions of ''G'' on ''H''. We say that a representation ρ is '''unitary''' if ρ(''g'') is a [[unitary operator]] for all ''g'' ∈ ''G''; i.e., <math>\langle gv,gw\rangle = \langle v,w\rangle</math> for all ''v'', ''w'' ∈ ''H''. (I.e. it is unitary if ρ : ''G'' → U(''H''). Notice how this generalises the special case of the one-dimensional Hilbert space, where U('''C''') is just the circle group.)
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| Given these definitions, we can state the second part of the Peter–Weyl theorem {{harv|Knapp|1986|loc=Theorem 1.14}}:
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| <blockquote>'''Peter-Weyl Theorem (Part II).''' Let ρ be a unitary representation of a compact group ''G'' on a complex Hilbert space ''H''. Then ''H'' splits into an orthogonal [[direct sum of representations|direct sum]] of irreducible finite-dimensional unitary representations of ''G''.</blockquote> | |
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| ==Decomposition of square-integrable functions==
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| To state the third and final part of the theorem, there is a natural Hilbert space over ''G'' consisting of [[square-integrable function]]s, [[Lp space|''L''<sup>2</sup>(''G'')]]; this makes sense because [[Haar measure]] exists on ''G''. Calling this Hilbert space ''H'', the group ''G'' has a [[unitary representation]] ρ on ''H'' by [[group action|acting]] on the left, via
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| :<math>\rho(h)f(g) = f(h^{-1}g).</math>
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| The final statement of the Peter–Weyl theorem {{harv|Knapp|1986|loc=Theorem 1.14}} gives an explicit [[orthonormal basis]] of ''L''<sup>2</sup>(''G''). Roughly it asserts that the matrix coefficients for ''G'', suitably renormalized, are an [[orthonormal basis]] of ''L''<sup>2</sup>(''G''). In particular, ''L''<sup>2</sup>(''G'') decomposes into an orthogonal direct sum of all the irreducible unitary representations, in which the multiplicity of each irreducible representation is equal to its degree (that is, the dimension of the underlying space of the representation). Thus,
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| :<math>L^2(G) = \underset{\pi\in\Sigma}{\widehat{\bigoplus}} E_\pi^{\oplus\dim E_\pi}</math>
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| where Σ denotes the set of (isomorphism classes of) irreducible unitary representations of ''G'', and the summation denotes the [[closure (topology)|closure]] of the direct sum of the total spaces ''E''<sub>π</sup> of the representations π.
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| More precisely, suppose that a representative π is chosen for each isomorphism class of irreducible unitary representation, and denote the collection of all such π by Σ. Let <math>\scriptstyle{u_{ij}^{(\pi)}}</math> be the matrix coefficients of π in an orthonormal basis, in other words
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| :<math>u^{(\pi)}_{ij}(g) = \langle \pi(g)e_i, e_j\rangle.</math>
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| for each ''g'' ∈ ''G''. Finally, let ''d''<sup>(π)</sup> be the degree of the representation π. The theorem now asserts that the set of functions
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| :<math>\left\{\sqrt{d^{(\pi)}}u^{(\pi)}_{ij}\mid\, \pi\in\Sigma,\,\, 1\le i,j\le d^{(\pi)}\right\}</math>
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| is an orthonormal basis of L<sup>2</sup>(''G'').
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| ==Consequences==
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| ===Structure of compact topological groups===
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| From the theorem, one can deduce a significant general structure theorem. Let ''G'' be a compact topological group, which we assume [[Hausdorff space|Hausdorff]]. For any finite-dimensional ''G''-invariant subspace ''V'' in ''L''<sup>2</sup>(''G''), where ''G'' [[group action|acts]] on the left, we consider the image of ''G'' in GL(''V''). It is closed, since ''G'' is compact, and a subgroup of the [[Lie group]] GL(''V''). It follows by a [[Cartan's theorem|theorem]] of [[Élie Cartan]] that the image of ''G'' is a Lie group also.
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| If we now take the limit (in the sense of [[category theory]]) over all such spaces ''V'', we get a result about ''G'' - because ''G'' acts faithfully on ''L''<sup>2</sup>(''G''). We can say that ''G'' is an ''inverse limit of Lie groups''. It may of course not itself be a Lie group: it may for example be a [[profinite group]].
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| ==See also==
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| * [[Pontryagin duality]]
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| ==References==
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| * {{citation|first1=F.|last1=Peter|first2=H.|last2=Weyl|title=Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe|journal=Math. Ann.|volume=97|year=1927|pages=737–755|doi=10.1007/BF01447892}}.
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| * {{citation|first1=R.S.|last1=Palais|first2=T.E.|last2=Stewart|title=The cohomology of differentiable transformation groups|journal=Amer. J. Math.|volume=83|issue=4|year=1961|pages=623–644|doi=10.2307/2372901|jstor=2372901|publisher=The Johns Hopkins University Press}}.
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| * {{citation|last=Knapp|first=Anthony|authorlink=Anthony Knapp|title=Representation theory of semisimple groups|publisher=Princeton University Press|year=1986|isbn=0-691-09089-0}}.
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| * {{citation|first=Daniel|last=Bump|title=Lie groups|publisher=Springer|year=2004|isbn=0-387-21154-3}}.
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| * {{citation|first=G.D.|last=Mostow|authorlink=George Mostow|title=Cohomology of topological groups and solvmanifolds|journal=Ann. Of Math.| volume=73|issue=1|year=1961|pages=20–48|doi=10.2307/1970281|jstor=1970281|publisher=Annals of Mathematics}}
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| * {{springer|title=Peter-Weyl theorem|id=p/p072440}}
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| {{DEFAULTSORT:Peter-Weyl theorem}}
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| [[Category:Unitary representation theory]]
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| [[Category:Topological groups]]
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| [[Category:Theorems in harmonic analysis]]
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| [[Category:Theorems in representation theory]]
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| [[Category:Theorems in group theory]]
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