Editing Unitary representation

{{Short description|Concept in mathematics}}
In [[mathematics]], a '''unitary representation''' of a [[Group (mathematics)|group]] ''G'' is a [[linear representation]] π of ''G'' on a complex [[Hilbert space]] ''V'' such that π(''g'') is a [[unitary operator]] for every ''g'' ∈ ''G''. The general theory is well-developed in the case that ''G'' is a [[locally compact]] ([[Hausdorff space|Hausdorff]]) [[topological group]] and the representations are [[strongly continuous]].

The theory has been widely applied in [[quantum mechanics]] since the 1920s, particularly influenced by [[Hermann 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, was [[George Mackey]].

==Context in harmonic analysis==

The theory of unitary representations of topological 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 by [[Pontryagin duality]]. In general, the unitary equivalence classes (see [[#Formal definitions|below]]) of [[irreducible representation|irreducible]] unitary representations of ''G'' make up its '''unitary dual'''.  This set can be identified with the [[spectrum of a C*-algebra|spectrum of the C*-algebra]] associated with ''G'' by the [[group ring|group C*-algebra]] construction.  This is a [[topological space]].

The general form of the [[Plancherel theorem]] tries to describe the [[regular representation]] of ''G'' on ''L''<sup>2</sup>(''G'') using a [[measure (mathematics)|measure]] on the unitary dual. For ''G'' abelian this is given by the Pontryagin duality theory.  For ''G'' [[Compact group|compact]], this is done by the [[Peter–Weyl theorem]]; in that case, the unitary dual is a [[discrete 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'',

:<math> \pi: G \rightarrow \operatorname{U}(H) </math>

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'').<ref>
Warner (1972)</ref> Smooth vectors are dense in ''H'' by a classical argument of [[Lars Gårding]], since convolution by smooth functions of [[Support (mathematics)#compact support|compact support]] yields smooth vectors. Analytic vectors are dense by a classical argument of [[Edward Nelson]], amplified by Roe Goodman, since vectors in the image of a  heat operator ''e''<sup>–tD</sup>, corresponding to an [[elliptic differential operator]] ''D'' in the [[universal enveloping algebra]] of ''G'', are analytic. Not only do smooth or analytic vectors form dense subspaces; but they also form common cores for the unbounded skew-adjoint operators corresponding to the elements of the [[Lie algebra]], in the sense of [[spectral theory]].<ref>Reed and Simon (1975)</ref>

Two unitary representations π<sub>1</sub>: ''G'' → U(''H''<sub>1</sub>), π<sub>2</sub>: ''G'' → U(''H''<sub>2</sub>) are said to be '''unitarily equivalent''' if there is a [[unitary transformation]] ''A'':''H''<sub>1</sub> → ''H''<sub>2</sub> such that π<sub>1</sub>(''g'') = ''A''<sup>*</sup> ∘ π<sub>2</sub>(''g'') ∘ ''A'' for all ''g'' in ''G''. When this holds, ''A'' is said to be an [[intertwining operator]] for the representations <math>(\pi_1,H_1),(\pi_2,H_2)</math>.<ref>[[Paul Sally]] (2013) ''Fundamentals of Mathematical Analysis'', [[American Mathematical Society]] [https://books.google.com/books?id=b05c370fLdsC&pg=PA234 pg. 234]</ref>

If <math>\pi</math> is a representation of a connected Lie group <math>G</math> on a ''finite-dimensional'' Hilbert space <math>H</math>, then <math>\pi</math> is unitary if and only if the associated Lie algebra representation <math>d\pi:\mathfrak{g}\rightarrow\mathrm{End}(H)</math> maps into the space of skew-self-adjoint operators on <math>H</math>.<ref>{{harvnb|Hall|2015}} Proposition 4.8</ref>

==Complete reducibility==

A unitary representation is [[Semisimple representation|completely reducible]], in the sense that for any closed [[invariant subspace]], the [[orthogonal 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 [[representations of a finite group|finite group]]s, and more generally for [[compact group]]s, by an averaging argument applied to an arbitrary hermitian structure (more specifically, a new inner product defined by an averaging argument over the old one, w.r.t which the representation is unitary).<ref>{{harvnb|Hall|2015}} Section 4.4</ref> For example, a natural proof of [[Maschke'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 group|reductive]] [[Lie group]]s. All [[Irreducible representation|irreducible]] unitary representations are [[Admissible representation|admissible]] (or rather their [[Harish-Chandra module]]s are), and the admissible representations are given by the [[Langlands 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 the quadratic form is [[Definite quadratic form|positive definite]]. For many reductive Lie groups this has been solved; see [[representation theory of SL2(R)]] and [[representation theory of the Lorentz group]] for examples.

==Notes==
{{reflist}}

==References==
* {{Citation| last=Hall|first=Brian C.|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn=978-3319134666}}
*{{citation|first=Michael |last=Reed|author1link = Michael C. Reed|author2link = Barry Simon|first2= Barry|last2= Simon|title=Methods of Modern Mathematical Physics, Vol. 2: Fourier Analysis, Self-Adjointness|publisher=Academic Press |year= 1975|isbn=0-12-585002-6}} 
*{{citation|title=Harmonic Analysis on Semi-simple Lie Groups I|first=Garth|last= Warner|year=1972|publisher=Springer-Verlag|isbn=0-387-05468-5}}

==See also==
*[[Induced representations]]
*[[Isotypical representation]]
*[[Representation theory of SL2(R)]]
*[[Representations of the Lorentz group]]
*[[Stone–von Neumann theorem]]
*[[Unitary representation of a star Lie superalgebra]] 
*[[Zonal spherical function]]

[[Category:Unitary representation theory| ]]