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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 case ''G'' is a [[locally compact]] (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 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 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 to ''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'') by means of 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 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; 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> | |||
==Complete reducibility== | |||
A unitary representation is [[Semisimple algebra|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. 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|first=Michael |last=Reed|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== | |||
*[[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]] | |||
[[Category:Unitary representation theory| ]] | |||
Revision as of 22:52, 1 January 2014
In mathematics, a unitary representation of a 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 case G is a locally compact (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 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 of irreducible unitary representations of G make 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 a topological space.
The general form of the Plancherel theorem tries to describe the regular representation of G on L2(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 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,
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).[1] Smooth vectors are dense in H by a classical argument of Lars Gårding, since convolution by smooth functions of 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–tD, 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; 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.[2]
Complete reducibility
A unitary representation is 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 finite groups, and more generally for compact groups, by an averaging argument applied to an arbitrary hermitian structure. 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 Lie groups. All irreducible unitary representations are admissible (or rather their Harish-Chandra modules 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 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
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References
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010