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Peter–Weyl theorem

In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G (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 Ferdinand Georg Frobenius and Issai Schur.

Let G be a compact group. The theorem has three parts. The first part states that the matrix coefficients of irreducible representations of G are dense in the space C(G) of continuous complex-valued functions on G, and thus also in the space L2(G) of square-integrable functions. The second part asserts the complete reducibility of unitary representations of G. The third part then asserts that the regular representation of G on L2(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 L2(G). In the case that G is the group of unit complex numbers, this last result is simply a standard result from Fourier series.

Consequences[edit]

Representation theory of connected compact Lie groups[edit]

The Peter–Weyl theorem—specifically the assertion that the characters form an orthonormal basis for the space of square-integrable class functions—plays a key role in the classification of the irreducible representations of a connected compact Lie group.[3] The argument also depends on the Weyl integral formula (for class functions) and the Weyl character formula.


An outline of the argument may be found here.

Linearity of compact Lie groups[edit]

One important consequence of the Peter–Weyl theorem is the following:[4]

Pontryagin duality

Peter, F.; Weyl, H. (1927), "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe", Math. Ann., 97: 737–755, :10.1007/BF01447892.

doi

Bump, Daniel (2004), Lie groups, Springer,  0-387-21154-3.

ISBN

Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,  978-3319134666.

ISBN

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Peter-Weyl theorem"

(1986), Representation theory of semisimple groups, Princeton University Press, ISBN 0-691-09089-0.

Knapp, Anthony

(2002), Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 140 (2nd ed.), Boston: Birkhäuser, ISBN 0-8176-4259-5.

Knapp, Anthony W.

(1961), "Cohomology of topological groups and solvmanifolds", Annals of Mathematics, 73 (1), Princeton University Press: 20–48, doi:10.2307/1970281, JSTOR 1970281

Mostow, George D.

; Stewart, T. E. (1961), "The cohomology of differentiable transformation groups", American Journal of Mathematics, 83 (4), The Johns Hopkins University Press: 623–644, doi:10.2307/2372901, JSTOR 2372901.

Palais, Richard S.