- Peter–Weyl theorem
In
mathematics , the Peter–Weyl theorem is a basic result in the theory ofharmonic analysis , applying totopological group s that arecompact , but are not necessarily abelian. It was initially proved byHermann Weyl , with his student Peter, in the setting of acompact topological group "G" harv|Peter|Weyl|1927. The theorem is a collection of results generalizing the significant facts about the decomposition of theregular representation of any finite group, as discovered byF. G. Frobenius andIssai Schur .Matrix coefficients
A
matrix coefficient of the group "G" is a complex-valued function φ on "G" given as the composition:phi = Lcirc pi
where π : "G" → GL("V") is a finite-dimensional (continuous)
group representation of "G", and "L" is alinear functional on the vector space ofendomorphism s of "V", 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.The first part of the Peter-Weyl theorem asserts (harvnb|Bump|2004|loc=§4.1; harvnb|Knapp|1986|loc=Theorem 1.12):
*The set of matrix coefficients of "G" is dense in the space of continuous complex functions C("G") on "G", equipped with the
uniform norm .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 amatrix group , then the result follows easily from the Stone-Weierstrass theorem harv|Knapp|1986|p=17. Conversely, it is a consequence of the subsequent conclusions of the theorem that any compactLie group is locally isomorphic to a matrix group harv|Knapp|1986|loc=Theorem 1.15.A corollary of this result is that the matrix coefficients of "G" are dense in L2("G").
Decomposition of a unitary representation
The second part of the theorem gives the existence of a decomposition of a
unitary representation of "G" into finite-dimensional representations. To state this part of the theorem we need first the idea of a unitary representation in aHilbert space . A Hilbert space is a completeinner product space . A representation ρ of "G" on a Hilbert space "H" is agroup homomorphism of "G" into the group GL("H") of bounded linear isomorphisms of "H" with itself with bounded inverses such that the map scriptstyle{(g,v)mapsto ho(g)v} is a continuous function:G imes H o H.
The representation ρ is unitary if ρ("g") is a
unitary operator for all "g" ∈ "G"; to wit,:langle ho(g)v, ho(g)w angle = langle v,w anglefor all "v", "w" ∈ "H".
The second part of the Peter-Weyl theorem asserts harv|Knapp|1986|loc=Theorem 1.14:
*Let ρ be a unitary representation of a compact group "G" on a Hilbert space "H". Then "H" splits into an orthogonal
direct sum of irreducible finite-dimensional unitary representations of "G".Decomposition of square-integrable functions
To state the third and final part of the theorem, there is a natural Hilbert space over "G" consisting of
square-integrable function s, "L"2("G"); this makes sense becauseHaar measure exists on "G". Calling this Hilbert space "H", the group "G" has aunitary representation ρ on "H" by acting on the left, via:ρ("h")f(g) = f(hg).
The final statement of the Peter-Weyl theorem harv|Knapp|1986|loc=Theorem 1.14 gives an explicit
orthonormal basis of L2("G"). Roughly it asserts that the matrix coefficients for "G", suitably renormalized, are anorthonormal basis of L2("G"). In particular, L2("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).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 scriptstyle{u_{ij}^{(pi) be the matrix coefficients of π in an orthonormal basis. Thus
:u^{(pi)}_{ij}(g) = langle pi(g)e_i, e_j angle.
for each "g" ∈ "G". Finally, let "d"(π) be the degree of the representation π. The theorem now asserts that the functions
:left{sqrt{d^{(pi)u^{(pi)}_{ij}mid, piinSigma,,, 1le i,jle d^{(pi)} ight}
is an orthonormal basis of L2("G").
Consequences
tructure of compact topological groups
From the theorem, one can deduce a significant general structure theorem. Let "G" be a compact topological group, which we assume Hausdorff. For any finite-dimensional "G"-invariant subspace "V" in "L"2("G"), where "G" 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 basic theorem (ofÉlie Cartan ) that the image of "G" is a Lie group also.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"2("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 aprofinite group .References
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