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Peter-Weyl Theorem


Let G be a Lie group and let rho be a group representation of G on C^n (for some natural number n), which is continuous in the sense that the function G×C^n->C^n defined by (g,v)|->rho(g)(v) is continuous. Then for each v in C^n and each alpha in (C^n)^*, the function G->C defined by g|->alpha(rho(g)(v)) is continuous. The vector space span of all such functions is called the space of representative functions.

The Peter-Weyl theorem says that, if G is compact, then

1. The representative functions are dense in the space of all continuous functions, with respect to the supremum norm;

2. The representative functions are dense in the space L^2(G) of all square-integrable functions, with respect to a Haar measure on G;

3. The vector space span of the characters of the irreducible continuous representations of G are dense in the space of all continuous functions from G into C which are constant on each conjugacy class of G, with respect to the supremum norm.

This theorem is easy to deduce from the Stone-Weierstrass theorem if it is assumed that G is a matrix group. On the other hand, it is a corollary of the Peter-Weyl theorem that every compact Lie group is isomorphic to some matrix group.


See also

Lie Group, Matrix Group, Norm, Stone-Weierstrass theorem, Supremum Norm

This entry contributed by José Carlos Santos

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References

Bröcker, T. and tom Dieck, T. Representations of Compact Lie Groups. New York: Springer-Verlag, 1985.Chevalley, C. Theory of Lie Groups. Princeton, NJ: Princeton University Press, 1999.Huang, J.-S. "The Peter-Weyl Theorem." §8.5 in Lectures on Representation Theory. Singapore: World Scientific, pp. 99-103, 1999.Knapp, A. W. "Group Representations and Harmonic Analysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.Peter, F. and Weyl, H. "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuerlichen Gruppe." Math. Ann. 97, 737-755, 1927.

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Peter-Weyl Theorem

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Santos, José Carlos. "Peter-Weyl Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Peter-WeylTheorem.html

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