Peter–Weyl theorem

In the mathematical subfield of harmonic analysis generalizes the theorem of Peter -Weyl, named after Hermann Weyl and his student Peter Fritz (1899-1949), the Fourier series for functions on arbitrary compact topological groups.

Representations on compact groups

Let be a compact topological group. For a complex Hilbert space name is a steady group homomorphism representation of the group, which is equipped with the weak operator topology. It can now show that every such a compact self-adjoint Vertauschungsoperator and thus as eigenspace of this operator is a finite, non-trivial invariant subspace of features. Therefore, every irreducible representation of a compact group is finite and each display can be a direct sum of such present is, it has a decomposition into irreducible representations.

Of particular interest is the left regular representation; this is defined by: where and with respect to the left-invariant square integrable on standardized measurement function, and hair is. One can show that for each such given by the above formula function is square- integrable again and that two almost everywhere equal functions reflects back on almost all the same functions, so total actually determines an operator on whose unitarity one can easily prove. Similarly, by and the two-sided presentation by defining the right regular representation.

For each presentation and is called matrix coefficient, a bounded continuous function (see Fourier - Stieltjes algebra).

Fourier transform

For all irreducible representations of you choose a system of representatives with respect to unitary equivalence. One of each representation corresponds to a Hilbert space representation of the Banach *-algebra with the convolution ( the so-called group algebra), so there is the equation for all. Since the Hair measure on a compact group is finite, is. For a function, the Fourier transform is now defined as, here is a picture of the orthogonal sum of the spaces of matrices, equipped with the Hilbert-Schmidt scalar product (this is in the compact case is always possible, since the representation spaces are finite ).

Set

The set of Peter -Weyl now states that the Fourier transform of a compact group up to some constant factors is unitary and constructed the inverse map. More precisely,

Unitary. The inverse map is given by

The track and denote the sum in the sense of unconditional convergence is to be understood.

Partial statements

Here are some partial statements are specified, which are sometimes used as evidence, and partly also in turn follow directly from the set of Peter -Weyl in the above form.

The rooms are pairwise orthogonal subspaces of, hence the subspaces are pairwise orthogonal and the operator is also unitary. If the family is an orthonormal basis of, the family of all dyadic products is an orthonormal basis of and thus orthonormal basis of. Are accordingly orthonormal bases given for each, the functions form an orthonormal basis of.

The presentation was defined as the outer tensor product with the contragredient representation, specifically:

The operator is now a Vertauschungsoperator between and, that is,

Which is equivalent to the constrained two-sided presentation on. If you choose fixed and normalized, so is the image of the operator

Invariant under the left regular representation, the unitary (with restriction of the image space ) operator

Is a Vertauschungsoperator between and. Thus, every irreducible representation of a compact group is equivalent to a partial representation of the left regular representation. The multiplicity of representation in the left regular representation, that is, how often this occurs in a decomposition into irreducibles is just equal to the dimension of the representation space. The orthogonal projection is given by a convolution. These results are completely analogous for the right regular representation by place and in the projection considers the inverse convolution.

Example

Be the circle group. Since is abelian, every irreducible representation is a character, ie a figure in the circle group itself These are just given by the functions. For and

And thus simple. This is nothing else than the known -th Fourier coefficient. Provides the set of Peter -Weyl (because the display is one-dimensional space, no further scaling needed ), the unitarity this transformation in the room as well as the inverse

710386
de