Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 022, 20 pages      arXiv:1102.4065

Conformally Equivariant Quantization - a Complete Classification

Jean-Philippe Michel
University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City, Luxembourg

Received July 29, 2011, in final form April 11, 2012; Published online April 15, 2012

Conformally equivariant quantization is a peculiar map between symbols of real weight δ and differential operators acting on tensor densities, whose real weights are designed by λ and λ+δ. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight δ. Later, Silhan has determined the critical values of δ for which unique existence is lost, and conjectured that for those values of δ existence is lost for a generic weight λ. We fully determine the cases of existence and uniqueness of the conformally equivariant quantization in terms of the values of δ and λ. Namely, (i) unique existence is lost if and only if there is a nontrivial conformally invariant differential operator on the space of symbols of weight δ, and (ii) in that case the conformally equivariant quantization exists only for a finite number of λ, corresponding to nontrivial conformally invariant differential operators on λ-densities. The assertion (i) is proved in the more general context of IFFT (or AHS) equivariant quantization.

Key words: quantization; (bi-)differential operators; conformal invariance; Lie algebra cohomology.

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  1. Beckmann R., Clerc J.L., Singular invariant trilinear forms and covariant (bi-)differential operators under the conformal group, J. Funct. Anal. 262 (2012), 4341-4376.
  2. Boe B.D., Collingwood D.H., A comparison theory for the structure of induced representations, J. Algebra 94 (1985), 511-545.
  3. Boe B.D., Collingwood D.H., A comparison theory for the structure of induced representations. II, Math. Z. 190 (1985), 1-11.
  4. Boniver F., Mathonet P., IFFT-equivariant quantizations, J. Geom. Phys. 56 (2006), 712-730, math.RT/0109032.
  5. Boniver F., Mathonet P., Maximal subalgebras of vector fields for equivariant quantizations, J. Math. Phys. 42 (2001), 582-589, math.DG/0009239.
  6. Cap A., Silhan J., Equivariant quantizations for AHS-structures, Adv. Math. 224 (2010), 1717-1734, arXiv:0904.3278.
  7. Duval C., Lecomte P., Ovsienko V., Conformally equivariant quantization: existence and uniqueness, Ann. Inst. Fourier (Grenoble) 49 (1999), 1999-2029, math.DG/9902032.
  8. Duval C., Ovsienko V., Conformally equivariant quantum Hamiltonians, Selecta Math. (N.S.) 7 (2001), 291-320, math.DG/9801122.
  9. Duval C., Ovsienko V., Projectively equivariant quantization and symbol calculus: noncommutative hypergeometric functions, Lett. Math. Phys. 57 (2001), 61-67, math.QA/0103096.
  10. Eastwood M., Slovák J., Semiholonomic Verma modules, J. Algebra 197 (1997), 424-448.
  11. Eastwood M.G., Rice J.W., Conformally invariant differential operators on Minkowski space and their curved analogues, Comm. Math. Phys. 109 (1987), 207-228, Erratum, Comm. Math. Phys. 144 (1992), 213.
  12. Fuks D.B., Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986.
  13. Kobayashi S., Nagano T., On filtered Lie algebras and geometric structures. I, J. Math. Mech. 13 (1964), 875-907.
  14. Kroeske J., Invariant bilinear differential pairings on parabolic geometries, Ph.D. thesis, University of Adelaide, 2008, arXiv:0904.3311.
  15. Kroeske J., Invariant differential pairings, Acta Math. Univ. Comenian. (N.S.) 77 (2008), 215-244, math.DG/0703866.
  16. Lecomte P.B.A., On the cohomology of sl(m+1,R) acting on differential operators and sl(m+1,R)-equivariant symbol, Indag. Math. (N.S.) 11 (2000), 95-114, math.DG/9801121.
  17. Lecomte P.B.A., Ovsienko V.Y., Projectively equivariant symbol calculus, Lett. Math. Phys. 49 (1999), 173-196, math.DG/9809061.
  18. Loubon Djounga S.E., Modules of third-order differential operators on a conformally flat manifold, J. Geom. Phys. 37 (2001), 251-261.
  19. Mathonet P., Radoux F., Cartan connections and natural and projectively equivariant quantizations, J. Lond. Math. Soc. (2) 76 (2007), 87-104, math.DG/0606556.
  20. Mathonet P., Radoux F., Existence of natural and conformally invariant quantizations of arbitrary symbols, J. Nonlinear Math. Phys. 17 (2010), 539-556, arXiv:0811.3710.
  21. Nikitin A.G., Prilipko A.I., Generalized Killing tensors and the symmetry of the Klein-Gordon-Fock equation, Preprint no. 90.23, Institute of Mathematics, Kyiv, 1990, 59 pages, math-ph/0506002.
  22. Ovsienko V., Redou P., Generalized transvectants-Rankin-Cohen brackets, Lett. Math. Phys. 63 (2003), 19-28, math.DG/0104232.
  23. Silhan J., Conformally invariant quantization - towards complete classification, arXiv:0903.4798.
  24. Weyl H., The classical groups. Their invariants and representations, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997.
  25. Wünsch V., On conformally invariant differential operators, Math. Nachr. 129 (1986), 269-281.

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