On établit l’existence et l’unicité d’un calcul symbolique et d’une quantification conformément équivariants sur une variété pseudo-riemannienne conformément plate , i.e. on met en évidence un isomorphisme canonique entre l’espace des polynômes sur et celui des opérateurs différentiels agissant sur les champs de densités tensorielles, vus tous les deux comme modules sur l’algèbre de Lie , où . Cette quantification existe pour des valeurs génériques des poids de densités; on calcule les valeurs critiques de ces poids, pour lesquelles un tel isomorphisme n’existe éventuellement pas. Dans le cas des demi-densités, on obtient ainsi un star-produit conformément équivariant.
We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-riemannian manifold . In other words, we establish a canonical isomorphism between the spaces of polynomials on and of differential operators on tensor densities over , both viewed as modules over the Lie algebra where . This quantization exists for generic values of the weights of the tensor densities and we compute the critical values of the weights yielding obstructions to the existence of such an isomorphism. In the particular case of half-densities, we obtain a conformally invariant star-product.
@article{AIF_1999__49_6_1999_0,
author = {Duval, Christian and Lecomte, Pierre and Ovsienko, Valentin},
title = {Conformally equivariant quantization: existence and uniqueness},
journal = {Annales de l'Institut Fourier},
pages = {1999--2029},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {49},
number = {6},
year = {1999},
doi = {10.5802/aif.1744},
zbl = {0932.53048},
mrnumber = {2000k:53081},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1744/}
}
TY - JOUR AU - Duval, Christian AU - Lecomte, Pierre AU - Ovsienko, Valentin TI - Conformally equivariant quantization: existence and uniqueness JO - Annales de l'Institut Fourier PY - 1999 SP - 1999 EP - 2029 VL - 49 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1744/ DO - 10.5802/aif.1744 LA - en ID - AIF_1999__49_6_1999_0 ER -
%0 Journal Article %A Duval, Christian %A Lecomte, Pierre %A Ovsienko, Valentin %T Conformally equivariant quantization: existence and uniqueness %J Annales de l'Institut Fourier %D 1999 %P 1999-2029 %V 49 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1744/ %R 10.5802/aif.1744 %G en %F AIF_1999__49_6_1999_0
Duval, Christian; Lecomte, Pierre; Ovsienko, Valentin. Conformally equivariant quantization: existence and uniqueness. Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1999-2029. doi : 10.5802/aif.1744. https://aif.centre-mersenne.org/articles/10.5802/aif.1744/
[1] , , , and , Deformation theory and quantization, Ann. Phys. (N.Y.), 111 (1978), 61-151. | MR | Zbl
[2] , Quantization, Math. USSR Izvestija, 8-5 (1974), 1109-1165. | MR | Zbl
[3] , Quantization in complex symmetric spaces, Math. USSR Izvestija, 9-2 (1976), 341-379. | MR | Zbl
[4] , Einstein Manifolds, Springer-Verlag, Berlin-Heidelberg, 1987. | MR | Zbl
[5] and , A remark about the Lie algebra of infinitesimal conformal transformations of the Euclidian space, math.DG/9901034. | Zbl
[6] , and , Automorphic pseudodifferential operators, Algebraic aspects of integrable systems, 17-47, Progr. Nonlinear Differential Equations Appl., 26, Birkhäuser Boston, Boston, MA, 1997. | MR | Zbl
[7] and , Formal deformation of the Poisson Lie algebra of a symplectic manifold and star-products, in Deformation Theory of Algebras and Structures And Applications, Kluver Acad. Pub., Dordrecht, 1988, and references therein. | Zbl
[8] and , Space of second order linear differential operators as a module over the Lie algebra of vector fields, Advances in Math., 132-2 (1997), 316-333. | MR | Zbl
[9] and , Conformally equivariant quantization, preprint CPT-98/P.3610, math. DG/9801122.
[10] , Deformation Quantization and Index Theory, Akademie Verlag, Berlin, 1996. | MR | Zbl
[11] , Sur la géométrie des opérateurs différentiels linéaires sur R, preprint CPT, 1997, P. 3472. | Zbl
[12] and , Space of linear differential operators on the real line as a module over the Lie algebra of vector fields, Int. Res. Math. Notes, 1996, No. 5, 235-251. | MR | Zbl
[13] , and , Lie algebras of vector fields in the real plane, Proc. London Math. Soc., 64 (1992), 339-368. | MR | Zbl
[14] , Elements of the theory of representations, Springer-Verlag, Berlin-New York, 1976. | MR | Zbl
[15] , Geometric Quantization, in Encyclopedia of Math. Sci. Vol. 4, Springer-Verlag, 1990. | Zbl
[16] , Closed algebras of differential operators, preprint 1996.
[17] , Quantization and Unitary Representations, in Lecture Notes in Math., Springer-Verlag, Vol. 170, 1970. | MR | Zbl
[18] , Symplectic Spinors, in Symposia Math., Vol. 14, London, Acad. Press, 1974. | MR | Zbl
[19] , On the cohomology of sl(m + 1, R) acting on differential operators and sl(m + 1, ℝ)-equivariant symbol, preprint Université de Liège, 1998.
[20] , Classification projective des espaces d'opérateurs différentiels agissant sur les densités, C.R.A.S., 328, Ser. 1 (1999). | MR | Zbl
[21] , and , Comparison of some modules of the Lie algebra of vector fields, Indag. Math., N.S., 7-4 (1996), 461-471. | MR | Zbl
[22] and , Projectively invariant symbol map and cohomology of vector fields Lie algebras intervening in quantization, dg-ga/9611006; Projectively invariant symbol calculus, math.DG/9809061.
[23] , Intertwining operators between some spaces of differential operators on a manifold, Comm. in Algebra (1999), to appear. | MR | Zbl
[24] and , Spinors and space-time, Vol. 2, Spinor and twistor methods in space-time geometry, Cambridge University Press, 1986. | MR | Zbl
[25] , A class of graded Lie algebras of vector fields and first order differential operators, J. Math. Phys., 35-12 (1994), 6338-6856. | MR | Zbl
[26] , Structure des systèmes dynamiques, Dunod, 1970, (c) 1969, Structure of Dynamical Systems. A Symplectic View of Physics, Birkhäuser, 1997. | Zbl
[27] , The Classical Groups, Princeton University Press, 1946.
[28] , Projective differential geometry of curves and ruled surfaces, Leipzig, Teubner, 1906. | JFM
Cité par Sources :
