Square Integrable Projective Representations and Square Integrable Representations Modulo a Relatively Central Subgroup

Aniello, Paolo

doi:10.1142/s0219887806001132

Cited by 27 publications

(31 citation statements)

References 24 publications

(50 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For a general proof of this uniqueness for G not unimodular, see [28] and also [4]. Because the U˙are induced from the subgroup f1; bg belonging to the kernel of , it is easily seen that d˙ .r/ D jrj .r/;…”

Section: The Duflo-moore Operatorsmentioning

confidence: 98%

Fourier analysis on the affine group, quantization and noncompact Connes geometries

Gayral¹,

Gracia‐Bondía²,

Várilly³

2008

J. Noncommut. Geom.

View full text Add to dashboard Cite

Abstract. We find the Stratonovich-Weyl quantizer for the nonunimodular affine group of the line. A noncommutative product of functions on the half-plane, underlying a noncompact spectral triple in the sense of Connes, is obtained from it. The corresponding Wigner functions reproduce the time-frequency distributions of signal processing. The same construction leads to scalar Fourier transformations on the affine group, simplifying and extending the Fourier transformation proposed by Kirillov.

show abstract

Section: The Duflo-moore Operatorsmentioning

confidence: 98%

Fourier analysis on the affine group, quantization and noncompact Connes geometries

Gayral¹,

Gracia‐Bondía²,

Várilly³

2008

J. Noncommut. Geom.

View full text Add to dashboard Cite

show abstract

“…Square integrable representations are ruled by the following fundamental result [7][8][9][10][11][12][13]:…”

Section: Functions Of This Form Allow Us To Define the Setmentioning

confidence: 99%

Discovering the manifold facets of a square integrable representation: from coherent states to open systems

Aniello

2019

J. Phys.: Conf. Ser.

Self Cite

View full text Add to dashboard Cite

Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary -or, in general, projective unitary -representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.

show abstract

“…The representation U is said to be square integrable if A(U ) = {0}. Square integrable projective representations are characterized by the following result-see [28]-which is a generalization of a classical theorem of Duflo and Moore [29] concerning unitary representations. …”

Section: Some Known Facts and Notationsmentioning

confidence: 99%

“…The ordinary wavelet transform arises in the special case where G is the one-dimensional affine group R R + * (see [30,31]); we will clarify this point in section 6. The isometry W ψ U intertwines the representation U with the left regular m-representation R m of G in L 2 (G), see [28], which is the projective representation (with multiplier m) defined by…”

Section: Remark 21mentioning

confidence: 99%

Snap Down Voltage of a Fast-Scanning Micromirror with Vertical Electrostatic Combdrives

Wada

Lee

Zappe

et al. 2004

Jpn. J. Appl. Phys.

View full text Add to dashboard Cite

Adopting a purely group-theoretical point of view, we consider the star product of functions which is associated, in a natural way, with a square integrable (in general, projective) representation of a locally compact group. Next, we show that for this (implicitly defined) star product, explicit formulas can be provided. Two significant examples are studied in detail: the group of translations on phase space and the one-dimensional affine group. The study of the first example leads to the Groenewold-Moyal star product. In the second example, the link with wavelet analysis is clarified.

show abstract

Square Integrable Projective Representations and Square Integrable Representations Modulo a Relatively Central Subgroup

Cited by 27 publications

References 24 publications

Fourier analysis on the affine group, quantization and noncompact Connes geometries

Fourier analysis on the affine group, quantization and noncompact Connes geometries

Discovering the manifold facets of a square integrable representation: from coherent states to open systems

Snap Down Voltage of a Fast-Scanning Micromirror with Vertical Electrostatic Combdrives

Contact Info

Product

Resources

About