Higher symmetries of the Laplacian via quantization

Jean-Philippe Michel[1]

  • [1] University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City, Grand Duchy of Luxembourg University of Liège, Sart-Tilman, 12 grande traverse, B-4000 Liège, Belgium

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 4, page 1581-1609
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Michel, Jean-Philippe. "Higher symmetries of the Laplacian via quantization." Annales de l’institut Fourier 64.4 (2014): 1581-1609. <http://eudml.org/doc/275610>.

@article{Michel2014,
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.},
affiliation = {University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City, Grand Duchy of Luxembourg University of Liège, Sart-Tilman, 12 grande traverse, B-4000 Liège, Belgium},
author = {Michel, Jean-Philippe},
journal = {Annales de l’institut Fourier},
keywords = {Symmetry algebra; Laplacian; Quantization; Conformal geometry; Minimal nilpotent orbit; Symplectic reduction; symmetry algebra; quantization; conformal geometry; minimal nilpotent orbit; symplectic reduction},
language = {eng},
number = {4},
pages = {1581-1609},
publisher = {Association des Annales de l’institut Fourier},
title = {Higher symmetries of the Laplacian via quantization},
url = {http://eudml.org/doc/275610},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Michel, Jean-Philippe
TI - Higher symmetries of the Laplacian via quantization
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 4
SP - 1581
EP - 1609
AB - 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.
LA - eng
KW - Symmetry algebra; Laplacian; Quantization; Conformal geometry; Minimal nilpotent orbit; Symplectic reduction; symmetry algebra; quantization; conformal geometry; minimal nilpotent orbit; symplectic reduction
UR - http://eudml.org/doc/275610
ER -

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