- Unitary representation
In
mathematics , a unitary representation of a group "G" is alinear representation π of "G" on a complexHilbert space "V" such that π("g") is aunitary operator for every "g" ∈ "G". The general theory is well-developed in case "G" is alocally compact (Hausdorff)topological group and the representations arestrongly continuous .The theory has been widely applied in
quantum mechanics since the1920s , particularly influenced byHermann Weyl 's 1928 book "Gruppentheorie und Quantenmechanik". One of the pioneers in constructing a general theory of unitary representations, for any group "G" rather than just for particular groups useful in applications, wasGeorge Mackey .Context in harmonic analysis
The theory of unitary representations of groups is closely connected with
harmonic analysis . In the case of an abelian group "G", a fairly complete picture of the representation theory of "G" is given byPontryagin duality . In general, the unitary equivalence classes of irreducible unitary representations of "G" makes up its unitary dual. This set can be identified with the spectrum of the C*-algebra associated to "G" by the group C*-algebra construction. This is atopological space .The general form of the
Plancherel theorem tries to describe the regular representation of "G" on "L"2("G") by means of a measure on the unitary dual. For "G" abelian this is given by the Pontryagin duality theory. For "G"compact , this is done by thePeter-Weyl theorem ; in that case the unitary dual is adiscrete space , and the measure attaches an atom to each point of mass equal to its degree.Formal definitions
Let "G" be a topological group. A strongly continuous unitary representation of "G" on a Hilbert space "H" is a group homomorphism from "G" into the unitary group of "H",
:
such that "g" → π("g") ξ is a norm continuous function for every ξ ∈ "H".
Note that if G is a
Lie group , the Hilbert space also admits underlying smooth and analytic structures. A vector ξ in "H" is said to be smooth or analytic if the map "g" → π("g") ξ is smooth or analytic (in the norm or weak topologies on "H"). [Warner (1972)] Smooth vectors are dense in "H" by a classical argument ofLars Gårding , since convolution by smooth functions of compact support yields smooth vectors. Analytic vectors are dense by a classical argument ofEdward Nelson , amplified by Roe Goodman, since vectors in the image of a heat operator "e"–tD, corresponding to anelliptic differential operator "D" in theuniversal enveloping algebra of "G", are analytic. Not only do smooth or analytic vectors form dense subspaces; they also form common cores for the unbounded skew-adjoint operators corresponding to the elements of theLie algebra , in the sense ofspectral theory . [ Reed and Simon (1975)]Complete reducibility
A unitary representation is
completely reducible , in the sense that for any closedinvariant subspace , theorthogonal complement is again a closed invariant subspace. This is at the level of an observation, but is a fundamental property. For example, it implies that finite dimensional unitary representations are always a direct sum of irreducible representations, in the algebraic sense.Since unitary representations are much easier to handle than the general case, it is natural to consider unitarizable representations, those that become unitary on the introduction of a suitable complex Hilbert space structure. This works very well for finite groups, and more generally for
compact group s, by an averaging argument applied to an arbitrary hermitian structure. For example, a natural proof ofMaschke's theorem is by this route.Unitarizability and the unitary dual question
In general, for non-compact groups, it is a more serious question which representations are unitarizable. One of the important unsolved problems in mathematics is the description of the unitary dual, the effective classification of irreducible unitary representations of all real
reductive Lie group s. All irreducible unitary representations are admissible (or rather theirHarish-Chandra module s are), and the admissible representations are given by theLanglands classification , and it is easy to tell which of them have a non-trivial invariant sesquilinear form. The problem is that it is in general hard to tell when this form is positive definite. For many reductive Lie groups this has been solved; seerepresentation theory of SL2(R) andrepresentation theory of the Lorentz group for examples.Notes
References
*citation|first=Michael |last=Reed|first2= Barry|last2= Simon|title=Methods of Modern Mathematical Physics, Vol. 2: Fourier Analysis, Self-Adjointness|publisher=Academic Press | id=ISBN 0125850026 |year= 1975
*citation|title=Harmonic Analysis on Semi-simple Lie Groups I|first=Garth|last= Warner|year=1972|publisher=Springer-Verlag|id=ISBN 0387054685ee also
*
Unitary representation of a star Lie superalgebra
*Representation theory of SL2(R)
*Representations of the Lorentz group
*Zonal spherical function
*Induced representations
*Stone-von Neumann theorem
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