Symmetric spaces and star representations

Bieliavsky, Pierre; Pevzner, M.

doi:10.1016/s0393-0440(01)00057-2

Cited by 4 publications

(3 citation statements)

References 12 publications

(10 reference statements)

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“…Semisimple symmetric spaces and star-representations [23,24] One of the possible approaches to the quantization of homogeneous spaces consists in adapting the Weyl calculus to the non-flat situation keeping working on the side of asymptotic expansions and looking for an alternative Moyal product that would respect the symmetries of the system that should be quantized. Such was the strategy developed in [23,24]. …”

Section: Star-representations and Their Applicationsmentioning

confidence: 99%

“…In order to construct a covariant star-product we need to identify representations ρ L and ρ R among those of the Lie algebra g. In [23] we considered the case where the strongly Hamiltonian system is a particular coadjoint orbit, to wit M = G/H is a causal symmetric space of Cayley type [54]. Such a symmetric space can be interpreted as the quotient of the conformal group G of a Euclidean Jordan algebra by the connected component of the identity of its structure group H. Using techniques of conformal groups introduced in the previous section we succeeded in the identification of the associated star-representations and constructed explicit intertwining operators between them and some of the holomorphic discrete series representations of G.…”

Section: )[[ ]] ×C ∞ (M)[[ ]] → C ∞ (M)[[ ]] Is C[[ ]]-Bilinearmentioning

confidence: 99%

“…The paper [24] is devoted to the construction of a convergent deformation quantization of the Poincaré upper half-plane G/K = Π. Starting from results described in [23], we studied the representation ad = (ρ L − ρ R ) of g acting by star-product's derivations on the formal power series on the onesheeted hyperboloid G/H and we showed that it can be restricted to the regular action of g on C ∞ c (Π). Using the Plancherel formula for L 2 (Π) we constructed an intertwining operator realizing the star-representation ad on the space of square integrable functions on Π.…”

Section: )[[ ]] ×C ∞ (M)[[ ]] → C ∞ (M)[[ ]] Is C[[ ]]-Bilinearmentioning

confidence: 99%

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Covariant quantization: spectral analysis versus deformation theory

Pevzner

2008

Jpn. J. Math.

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Formal deformation or rather symbolic calculus? To which extent do these approaches complete each other in the study of symmetry-preserving quantization procedures for homogeneous spaces? The representation theory of underlying Lie groups shows that the answer is much more delicate than initially thought and that it cannot be always reduced to asymptotic expansions with respect to some Planck's constant. The main goal of this survey is to give hints regarding the aims of each approach and, on the domain where these intersect, to compare the answers they lead to.

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Section: Star-representations and Their Applicationsmentioning

confidence: 99%

Section: )[[ ]] ×C ∞ (M)[[ ]] → C ∞ (M)[[ ]] Is C[[ ]]-Bilinearmentioning

confidence: 99%

Section: )[[ ]] ×C ∞ (M)[[ ]] → C ∞ (M)[[ ]] Is C[[ ]]-Bilinearmentioning

confidence: 99%

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Covariant quantization: spectral analysis versus deformation theory

Pevzner

2008

Jpn. J. Math.

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Symmetric spaces and star representations III. The Poincaré disc

Bieliavsky

Pevzner

2004

Noncommutative Harmonic Analysis

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This article is a contribution to the domain of (convergent) deformation quantization of symmetric spaces by use of Lie groups representation theory. We realize the regular representation of SL(2, R) on the space of smooth functions on the Poincaré disc as a sub-representation of SL(2, R) in the Weyl-Moyal star product algebra on R 2 . We indicate how it is possible to extend our construction to the general case of a Hermitian symmetric space of tube type.2000 Mathematics Subject Classification: 53D55, 22E45.

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Simple space-time symmetries: Generalizing conformal field theory

Mack

Riese

2007

Journal of Mathematical Physics

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We study simple space-time symmetry groups G which act on a space-time manifold Å = G H which admits a G-invariant global causal structure. We classify pairs (G, Å) which share the following additional properties of conformal field theory: 1) The stability subgroup H of o ∈ Å is the identity component of a parabolic subgroup of G, implying factorization H = MAN − , where M generalizes Lorentz transformations, A dilatations, and N − special conformal transformations. 2) special conformal transformations ξ ∈ N − act trivially on tangent vectors v ∈ T o Å. The allowed simple Lie groups G are the universal coverings of SU (m, m), SO(2, D), Sp(l, Ê), SO * (4n) and E 7(−25) andH are particular maximal parabolic subgroups. They coincide with the groups of fractional linear transformations of Euklidean Jordan algebras whose use as generalizations of Minkowski space time was advocated by Günaydin. All these groups G admit positive energy representations. It will also be shown that the classical conformal groups SO(2, D) are the only allowed groups which possess a time reflection automorphism T ; in all other cases space-time has an intrinsic chiral structure.

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Symmetric spaces and star representations

Cited by 4 publications

References 12 publications

Covariant quantization: spectral analysis versus deformation theory

Covariant quantization: spectral analysis versus deformation theory

Symmetric spaces and star representations III. The Poincaré disc

Simple space-time symmetries: Generalizing conformal field theory

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