Higher symmetries of the Laplacian via quantization

[Symétries supérieures du laplacien par quantification]

Michel, Jean-Philippe

Annales de l'Institut Fourier, Tome 64 (2014), p. 1581-1609 / Harvested from Numdam

Résumé

Nous développons une nouvelle approche, basée sur des méthodes de quantification, pour étudier les symétries supérieures d’opérateurs différentiels invariants. Nous traitons ici le cas des puissances conformes du laplacien sur une variété conformément plate et retrouvons les résultats de Eastwood, Leistner, Gover et Šilhan. En particulier, la quantifciation conformément équivariante établit une correspondence entre l’algèbre des symétries hamiltoniennes du flot géodésique nul et l’algèbre des symétries supérieures du laplacien conforme. Via une réduction symplectique, ceci conduit à une quantification de l’orbite nilpotente minimale du groupe conforme. La star-déformation de son algèbre de fonctions régulières est isomorphe à l’algèbre des symétries supérieures du laplacien conforme. Les deux s’identifient au quotient de l’algèbre enveloppante de l’algèbre de Lie conforme par l’idéal de Joseph.

Abstract

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.

Détail

Publié le : 2014-01-01

MR 3329674 | Zbl 06387318

DOI : https://doi.org/10.5802/aif.2891
Classification:  58J10,  53A30,  70S10,  17B08,  53D20,  53D55
Mots clés: algèbre de symétries, laplacien, quantification, géometrie conforme, orbite nilpotente minimale, réduction symplectique.

Bibliographie

[1] Arnal, D.; Benamor, H.; Cahen, B. Algebraic deformation program on minimal nilpotent orbit, Lett. Math. Phys., Tome 30 (1994) no. 3, pp. 241-250 | MR 1267005 | Zbl 0805.17009

[2] Astashkevich, A.; Brylinski, R. Non-local equivariant star product on the minimal nilpotent orbit, Adv. Math., Tome 171 (2002) no. 1, pp. 86-102 | Article | MR 1933385 | Zbl 1010.22021

[3] Balleier, C.; Wurzbacher, T. On the geometry and quantization of symplectic Howe pairs, Math. Zeit., Tome 271 (2012), pp. 577-591 | Article | MR 2917159 | Zbl 1252.53097

[4] Bekaert, X.; Grigoriev, M. Manifestly conformal descriptions and higher symmetries of bosonic singletons, SIGMA, Tome 6 (2010) (Paper 038) | MR 2647317 | Zbl 1241.70049

[5] Bekaert, Xavier; Meunier, Elisa; Moroz, Sergej Symmetries and currents of the ideal and unitary fermi gases, JHEP, Tome 2012 (2012) no. 2 (Article:113) | Article

[6] Binegar, B.; Zierau, R. Unitarization of a singular representation of ${\rm SO}(p,q)$, Comm. Math. Phys., Tome 138 (1991) no. 2, pp. 245-258 | Article | MR 1108044 | Zbl 0748.22009

[7] Boe, B. D.; Collingwood, D. H. A comparison theory for the structure of induced representations, J. Algebra, Tome 94 (1985) no. 2, pp. 511-545 | Article | MR 792968 | Zbl 0606.17007

[8] Boe, B. D.; Collingwood, D. H. A comparison theory for the structure of induced representations. II, Math. Z., Tome 190 (1985) no. 1, pp. 1-11 | Article | MR 793343 | Zbl 0562.17003

[9] Boniver, F.; Mathonet, P. IFFT-equivariant quantizations, J. Geom. Phys., Tome 56 (2006) no. 4, pp. 712-730 | Article | MR 2199289 | 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., Tome 60 (1976), pp. 35-80 | MR 393791 | Zbl 0314.33011

[11] Calderbank, David M. J.; Diemer, Tammo Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math., Tome 537 (2001), pp. 67-103 | MR 1856258 | Zbl 0985.58002

[12] Čap, A.; Šilhan, J. Equivariant quantizations for AHS-structures, Adv. Math., Tome 224 (2010) no. 4, pp. 1717-1734 | Article | MR 2646309 | Zbl 1193.53034

[13] Cordani, Bruno Conformal regularization of the Kepler problem, Comm. Math. Phys., Tome 103 (1986) no. 3, pp. 403-413 | Article | MR 832916 | Zbl 0599.70014

[14] Dairbekov, N.; Sharafutdinov, V. On conformal killing symmetric tensor fields on Riemannian manifolds, Sib. Adv. Math., Tome 21 (2011), pp. 1-41 | Article | MR 2682769 | Zbl 1249.53050

[15] Dixmier, Jacques Algèbres enveloppantes, Gauthiers-Villars, Paris (1974) | MR 498737 | Zbl 0308.17007

[16] Duval, C.; El Gradechi, A. M.; Yu. Ovsienko, V. Projectively and conformally invariant star-products, Comm. Math. Phys., Tome 244 (2004) no. 1, pp. 3-27 | Article | MR 2029948 | Zbl 1048.53063

[17] Duval, C.; Lecomte, P. B. A.; Yu. Ovsienko, V. Conformally equivariant quantization: existence and uniqueness, Ann. Inst. Fourier (Grenoble), Tome 49 (1999) no. 6, pp. 1999-2029 | Article | Numdam | MR 1738073 | Zbl 0932.53048

[18] Duval, C.; Yu. Ovsienko, V. Conformally equivariant quantum Hamiltonians, Selecta Math. (N.S.), Tome 7 (2001) no. 3, pp. 291-320 | Article | MR 1868301 | Zbl 1018.53041

[19] Eastwood, M. G. Higher symmetries of the Laplacian, Ann. of Math. (2), Tome 161 (2005) no. 3, pp. 1645-1665 | Article | MR 2180410 | Zbl 1091.53020

[20] Eastwood, M. G.; Leistner, T. Higher symmetries of the square of the Laplacian, Symmetries and overdetermined systems of partial differential equations, Springer, New York (IMA Vol. Math. Appl.) Tome 144 (2008), pp. 319-338 | MR 2384717 | Zbl 1137.58014

[21] Eastwood, M. G.; Rice, J. W. Conformally invariant differential operators on Minkowski space and their curved analogues, Comm. Math. Phys., Tome 109 (1987) no. 2, pp. 207-228 (Erratum Comm. Math. Phys., 144 (1992) no. 2, p. 213.) | Article | MR 880414 | 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., Tome 57 (2007) no. 12, pp. 2539-2546 | Article | MR 2369839 | Zbl 1161.17006

[23] Fioresi, R.; Lledó, M. A. On the deformation quantization of coadjoint orbits of semisimple groups, Pacific J. Math., Tome 198 (2001) no. 2, pp. 411-436 | Article | MR 1835516 | Zbl 1053.53057

[24] Gover, A. R.; Peterson, L. J. Conformally invariant powers of the Laplacian, $Q$-curvature, and tractor calculus, Comm. Math. Phys., Tome 235 (2003) no. 2, pp. 339-378 | Article | MR 1969732 | 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., Tome 53 (2012) no. 3 (Article 032301) | Article | MR 2798213 | Zbl 1274.35063

[26] Howe, R. Remarks on classical invariant theory, Trans. Amer. Math. Soc., Tome 313 (1989) no. 2, pp. 539-570 (Erratum Trans. Amer. Math. Soc., 318 (1990) no. 2, p. 823) | Article | MR 986027 | 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), Tome 9 (1976) no. 1, pp. 1-29 | Numdam | MR 404366 | Zbl 0346.17008

[28] Kobayashi, T.; Ørsted, B. Analysis on the minimal representation of $\mathrm{O}(p,q)$. I. Realization via conformal geometry, Adv. Math., Tome 180 (2003) no. 2, pp. 486-512 | Article | MR 2020550 | Zbl 1046.22004

[29] Kobayashi, T.; Ørsted, B. Analysis on the minimal representation of $\mathrm{O}(p,q)$. III. Ultrahyperbolic equations on $\mathbb{R}^{p-1,q-1}$, Adv. Math., Tome 180 (2003) no. 2, pp. 551-595 | Article | MR 2020552 | 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., Tome 124 (2000) no. 1, pp. 95-110 | Article | MR 1797655 | Zbl 0968.17007

[31] Lepowsky, J. A generalization of the Bernstein–Gelfand–Gelfand resolution, J. Algebra, Tome 49 (1977), pp. 496-511 | Article | MR 476813 | Zbl 0381.17006

[32] Loubon Djounga, S. E. Conformally invariant quantization at order three, Lett. Math. Phys., Tome 64 (2003) no. 3, pp. 203-212 | Article | MR 2009258 | Zbl 1056.53059

[33] Michel, J.-Ph. Conformally equivariant quantization–a complete classification, SIGMA, Tome 8 (2012), pp. 20 pp (paper 022) | MR 2942817 | Zbl 1243.53133

[34] Michel, J.-Ph.; Radoux, F.; Šilhan, J. Second order symmetries of the conformal Laplacian, SIGMA, Tome 10 (2014) (Article 016) | MR 3210619 | Zbl 1288.58014

[35] Nikitin, A. G.; Prilipko, A. I. Generalized Killing tensors and the symmetry of the Klein-Gordon-Fock equation (1990) | MR 1107804 | Zbl 0736.35091

[36] Ortega, J.-P.; Ratiu, T.S. Momentum maps and Hamiltonian reduction, Birkhäuser, Basel, Progress in Mathematics, Tome 222 (2004) | MR 2021152 | Zbl 1241.53069

[37] Radoux, F. An explicit formula for the natural and conformally invariant quantization, Lett. Math. Phys., Tome 89 (2009) no. 3, pp. 249-263 | Article | MR 2551182 | Zbl 1179.53014

[38] Šilhan, J. Conformally invariant quantization - towards complete classification, Differ. geom. appl., Tome 33, Supplement(0) (2014), pp. 162-174 (The Interaction of Geometry and Representation Theory. Exploring new frontiers) | Article | MR 3159956 | Zbl 1282.53077

[39] Somberg, P. Deformations of quadratic algebras, the Joseph ideal for classical Lie algebras, and special tensors, Symmetries and overdetermined systems of partial differential equations, Springer, New York (IMA Vol. Math. Appl.) Tome 144 (2008), pp. 527-536 | MR 2384730 | Zbl 1182.17002

[40] Vlasáková, Z. Symmetries of CR sub-Laplacian (2012) (arXiv:1201.6219)

[41] Weyl, H. The classical groups, Princeton University Press, Princeton, NJ, Princeton Landmarks in Mathematics (1997) (Their invariants and representations, Fifteenth printing, Princeton Paperbacks) | MR 1488158

[42] Wolf, J. A. Representations associated to minimal co-adjoint orbits, Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977), Springer, Berlin (Lecture Notes in Math.) Tome 676 (1978), pp. 329-349 | MR 519619 | Zbl 0388.22008

[43] Wünsch, V. On conformally invariant differential operators, Math. Nachr., Tome 129 (1986), pp. 269-281 | Article | MR 864639 | Zbl 0619.53008