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Higher symmetries of the Laplacian via quantization. (Symétries supérieures du laplacien par quantification.) (English. French summary) Zbl 1307.53076

Summary: We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of Eastwood, Leistner, Gover and Šilhan. In particular, conformally equivariant quantization establishes a correspondence between the algebra of Hamiltonian symmetries of the null geodesic flow and the algebra of higher symmetries of the conformal Laplacian. Combined with a symplectic reduction, this leads to a quantization of the minimal nilpotent coadjoint orbit of the conformal group. The star-deformation of its algebra of regular functions is isomorphic to the algebra of higher symmetries of the conformal Laplacian. Both identify with the quotient of the universal envelopping algebra by the Joseph ideal.

MSC:

53D55 Deformation quantization, star products
58J10 Differential complexes
53A30 Conformal differential geometry (MSC2010)
70S10 Symmetries and conservation laws in mechanics of particles and systems
17B08 Coadjoint orbits; nilpotent varieties
53D20 Momentum maps; symplectic reduction
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[1] Arnal, D.; Benamor, H.; Cahen, B., Algebraic deformation program on minimal nilpotent orbit, Lett. Math. Phys., 30, 3, 241-250 (1994) · Zbl 0805.17009
[2] Astashkevich, A.; Brylinski, R., Non-local equivariant star product on the minimal nilpotent orbit, Adv. Math., 171, 1, 86-102 (2002) · Zbl 1010.22021
[3] Balleier, C.; Wurzbacher, T., On the geometry and quantization of symplectic Howe pairs, Math. Zeit., 271, 577-591 (2012) · Zbl 1252.53097
[4] Bekaert, X.; Grigoriev, M., Manifestly conformal descriptions and higher symmetries of bosonic singletons, SIGMA, 6 (2010) · Zbl 1241.70049
[5] Bekaert, Xavier; Meunier, Elisa; Moroz, Sergej, Symmetries and currents of the ideal and unitary fermi gases, JHEP, 2012, 2 (2012) · Zbl 1309.81177
[6] Binegar, B.; Zierau, R., Unitarization of a singular representation of \({\rm SO}(p,q)\), Comm. Math. Phys., 138, 2, 245-258 (1991) · Zbl 0748.22009
[7] Boe, B. D.; Collingwood, D. H., A comparison theory for the structure of induced representations, J. Algebra, 94, 2, 511-545 (1985) · Zbl 0606.17007
[8] Boe, B. D.; Collingwood, D. H., A comparison theory for the structure of induced representations. II, Math. Z., 190, 1, 1-11 (1985) · Zbl 0562.17003
[9] Boniver, F.; Mathonet, P., IFFT-equivariant quantizations, J. Geom. Phys., 56, 4, 712-730 (2006) · Zbl 1145.53069
[10] Boyer, C. P.; Kalnins, E. G.; Miller, Jr., W., Symmetry and separation of variables for the Helmholtz and Laplace equations, Nagoya Math. J., 60, 35-80 (1976) · Zbl 0314.33011
[11] Calderbank, David M. J.; Diemer, Tammo, Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math., 537, 67-103 (2001) · Zbl 0985.58002
[12] Čap, A.; Šilhan, J., Equivariant quantizations for AHS-structures, Adv. Math., 224, 4, 1717-1734 (2010) · Zbl 1193.53034
[13] Cordani, Bruno, Conformal regularization of the Kepler problem, Comm. Math. Phys., 103, 3, 403-413 (1986) · Zbl 0599.70014
[14] Dairbekov, N.; Sharafutdinov, V., On conformal killing symmetric tensor fields on Riemannian manifolds, Sib. Adv. Math., 21, 1-41 (2011) · Zbl 1249.53050
[15] Dixmier, Jacques, Algèbres enveloppantes (1974) · Zbl 0308.17007
[16] Duval, C.; El Gradechi, A. M.; Yu. Ovsienko, V., Projectively and conformally invariant star-products, Comm. Math. Phys., 244, 1, 3-27 (2004) · Zbl 1048.53063
[17] Duval, C.; Lecomte, P. B. A.; Yu. Ovsienko, V., Conformally equivariant quantization: existence and uniqueness, Ann. Inst. Fourier (Grenoble), 49, 6, 1999-2029 (1999) · Zbl 0932.53048
[18] Duval, C.; Yu. Ovsienko, V., Conformally equivariant quantum Hamiltonians, Selecta Math. (N.S.), 7, 3, 291-320 (2001) · Zbl 1018.53041
[19] Eastwood, M. G., Higher symmetries of the Laplacian, Ann. of Math. (2), 161, 3, 1645-1665 (2005) · Zbl 1091.53020
[20] Eastwood, M. G.; Leistner, T., Symmetries and overdetermined systems of partial differential equations, 144, 319-338 (2008) · Zbl 1137.58014
[21] Eastwood, M. G.; Rice, J. W., Conformally invariant differential operators on Minkowski space and their curved analogues, Comm. Math. Phys., 109, 2, 207-228 (1987) · Zbl 0659.53047
[22] Eastwood, M. G.; Somberg, P.; Souček, V., Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras, J. Geom. Phys., 57, 12, 2539-2546 (2007) · Zbl 1161.17006
[23] Fioresi, R.; Lledó, M. A., On the deformation quantization of coadjoint orbits of semisimple groups, Pacific J. Math., 198, 2, 411-436 (2001) · Zbl 1053.53057
[24] Gover, A. R.; Peterson, L. J., Conformally invariant powers of the Laplacian, \(Q\)-curvature, and tractor calculus, Comm. Math. Phys., 235, 2, 339-378 (2003) · Zbl 1022.58014
[25] Gover, A. R.; Šilhan, J., Higher symmetries of the conformal powers of the Laplacian on conformally flat manifolds, J. Math Phys., 53, 3 (2012) · Zbl 1274.35063
[26] Howe, R., Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313, 2, 539-570 (1989) · Zbl 0674.15021
[27] Joseph, A., The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. École Norm. Sup. (4), 9, 1, 1-29 (1976) · Zbl 0346.17008
[28] Kobayashi, T.; Ørsted, B., Analysis on the minimal representation of \(\text{O}(p,q)\). I. Realization via conformal geometry, Adv. Math., 180, 2, 486-512 (2003) · Zbl 1046.22004
[29] Kobayashi, T.; Ørsted, B., Analysis on the minimal representation of \(\text{O}(p,q)\). III. Ultrahyperbolic equations on \(\mathbb{R}^{p-1,q-1} \), Adv. Math., 180, 2, 551-595 (2003) · Zbl 1039.22005
[30] Lecomte, P. B. A.; Yu. Ovsienko, V., Cohomology of the vector fields Lie algebra and modules of differential operators on a smooth manifold, Compositio Math., 124, 1, 95-110 (2000) · Zbl 0968.17007
[31] Lepowsky, J., A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra, 49, 496-511 (1977) · Zbl 0381.17006
[32] Loubon Djounga, S. E., Conformally invariant quantization at order three, Lett. Math. Phys., 64, 3, 203-212 (2003) · Zbl 1056.53059
[33] Michel, J.-Ph., Conformally equivariant quantization-a complete classification, SIGMA, 8 (2012) · Zbl 1243.53133
[34] Michel, J.-Ph.; Radoux, F.; Šilhan, J., Second order symmetries of the conformal Laplacian, SIGMA, 10 (2014) · Zbl 1288.58014
[35] Nikitin, A. G.; Prilipko, A. I., Generalized Killing tensors and the symmetry of the Klein-Gordon-Fock equation (1990) · Zbl 0736.35091
[36] Ortega, J.-P.; Ratiu, T. S., Momentum maps and Hamiltonian reduction, 222 (2004) · Zbl 1241.53069
[37] Radoux, F., An explicit formula for the natural and conformally invariant quantization, Lett. Math. Phys., 89, 3, 249-263 (2009) · Zbl 1179.53014
[38] Šilhan, J., Conformally invariant quantization - towards complete classification, Differ. geom. appl., 33, Supplement(0), 162-174 (2014) · Zbl 1282.53077
[39] Somberg, P., Symmetries and overdetermined systems of partial differential equations, 144, 527-536 (2008) · Zbl 1182.17002
[40] Vlasáková, Z., Symmetries of CR sub-Laplacian (2012)
[41] Weyl, H., The classical groups (1997) · JFM 65.0058.02
[42] Wolf, J. A., Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977), 676, 329-349 (1978) · Zbl 0388.22008
[43] Wünsch, V., On conformally invariant differential operators, Math. Nachr., 129, 269-281 (1986) · Zbl 0619.53008
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