Skip to main content
SpringerLink
Log in

Projectively Equivariant Quantizations over the Superspace \({\mathbb{R}^{p|q}}\)

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra \({\mathfrak{pgl}(p+1|q)}\) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of \({\mathfrak{pgl}(n|n)\not\cong \mathfrak{sl}(n|n)}\)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berezin, F.A.: Introduction to superanalysis. Mathematical Physics and Applied Mathematics, vol. 9. D. Reidel Publishing Co., Dordrecht (1987). Edited and with a foreword by A. A. Kirillov, with an appendix by V. I. Ogievetsky, translated from the Russian by J. Niederle and R. Kotecký, translation edited by D. Leites

  2. Bernstein, J.N., Leites, D.A.: Invariant differential operators and irreducible representations of Lie superalgebras of vector fields (Russian). Serdica 7(4), 320–334 (1981). English translation: Selecta Math. Soviet. 1(2), 143–160 (selected translations, 1981)

    Google Scholar 

  3. Boniver F., Hansoul S., Mathonet P., Poncin N.: Equivariant symbol calculus for differential operators acting on forms. Lett. Math. Phys. 62(3), 219–232 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boniver F., Mathonet P.: IFFT-equivariant quantizations. J. Geom. Phys. 56(4), 712–730 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Bordemann, M.: Sur l’existence d’une prescription d’ordre naturelle projectivement invariante. arXiv:math.DG/0208171

  6. Bouarroudj S.: Projectively equivariant quantization map. Lett. Math. Phys. 51(4), 265–274 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bouarroudj S.: Formula for the projectively invariant quantization on degree three. C. R. Acad. Sci. Paris Sér. I Math. 333(4), 343–346 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Čap A., Šilhan J.: Equivariant quantizations for AHS-structures. Adv. Math. 224(4), 1717–1734 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duval C., Lecomte P., Ovsienko V.: Conformally equivariant quantization: existence and uniqueness. Ann. Inst. Fourier (Grenoble) 49(6), 1999–2029 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Duval C., Ovsienko V.: Conformally equivariant quantum Hamiltonians. Selecta Math. (N.S.) 7(3), 291–320 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Duval C., Ovsienko V.: Projectively equivariant quantization and symbol calculus: noncommutative hypergeometric functions. Lett. Math. Phys. 57(1), 61–67 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fox D.J.F.: Projectively invariant star products. IMRP Int. Math. Res. Pap. 9, 461–510 (2005)

    Article  Google Scholar 

  13. Gargoubi H., Mellouli N., Ovsienko V.: Differential operators on supercircle: conformally equivariant quantization and symbol calculus. Lett. Math. Phys. 79(1), 51–65 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. George, J.: Projective connections and Schwarzian derivatives for supermanifolds, and Batalin-Vilkovisky operators. arXiv:0909.5419v1

  15. Grozman, P., Leites, D., Shchepochkina, I.: Invariant operators on supermanifolds and standard models. In: Multiple Facets of Quantization and Supersymmetry, pp. 508–555. World Science Publ., River Edge (2002). arXiv:math/0202193v2

  16. Hansoul S.: Projectively equivariant quantization for differential operators acting on forms. Lett. Math. Phys. 70(2), 141–153 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Hansoul S.: Existence of natural and projectively equivariant quantizations. Adv. Math. 214(2), 832–864 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kac, V.: Representations of classical Lie superalgebras. In: Differential Geometrical Methods in Mathematical Physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977). Lecture Notes in Math., vol. 676, pp. 597–626. Springer, Berlin (1978)

  19. Kac V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)

    Article  MATH  Google Scholar 

  20. Kac V.G.: A sketch of Lie superalgebra theory. Commun. Math. Phys. 53(1), 31–64 (1977)

    Article  ADS  MATH  Google Scholar 

  21. Kaplansky, I.: Graded Lie algebras I. http://justpasha.org/math/links/subj/lie/kaplansky

  22. Lecomte P.B.A., Ovsienko V.Yu.: Projectively equivariant symbol calculus. Lett. Math. Phys. 49(3), 173–196 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lecomte P.B.A.: Classification projective des espaces d’opérateurs différentiels agissant sur les densités. C. R. Acad. Sci. Paris Sér. I Math. 328(4), 287–290 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Lecomte, P.B.A.: Towards projectively equivariant quantization. Progr. Theoret. Phys. Suppl. 144, 125–132 (2001). Noncommutative geometry and string theory (Yokohama, 2001)

    Google Scholar 

  25. Loubon Djounga S.E.: Conformally invariant quantization at order three. Lett. Math. Phys. 64(3), 203–212 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mathonet P., Radoux F.: Existence of natural and conformally invariant quantizations of arbitrary symbols. J. Nonlinear Math. Phys. 17(4), 539–556 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Mathonet P., Radoux F.: Cartan connections and natural and projectively equivariant quantizations. J. Lond. Math. Soc. (2) 76(1), 87–104 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mathonet P., Radoux F.: On natural and conformally equivariant quantizations. J. Lond. Math. Soc. (2) 80(1), 256–272 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mathonet P., Radoux F.: Natural and projectively equivariant quantiations by means of Cartan connections. Lett. Math. Phys. 72(3), 183–196 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Mellouli, N.: Second-order conformally equivariant quantization in dimension 1|2. SIGMA 5(111) (2009)

  31. Michel, J.-P.: Quantification conformément équivariante des fibrés supercotangents. Thèse de Doctorat, Université Aix-Marseille II. http://tel.archives-ouvertes.fr/tel-00425576/fr/, (2009)

  32. Musson I.M.: On the center of the enveloping algebra of a classical simple Lie superalgebra. J. Algebra 193(1), 75–101 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pinczon G.: The enveloping algebra of the Lie superalgebra osp(1,2). J. Algebra 132(1), 219–242 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  34. Radoux F.: Explicit formula for the natural and projectively equivariant quantization. Lett. Math. Phys. 78(2), 173–188 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Radoux F.: Non-uniqueness of the natural and projectively equivariant quantization. J. Geom. Phys. 58(2), 253–258 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Radoux F.: An explicit formula for the natural and conformally invariant quantization. Lett. Math. Phys. 89(3), 249–263 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sergeev, A.N.: The invariant polynomials on simple Lie superalgebras. Represent. Theory 3, 250–280 (electronic, 1999)

    Google Scholar 

  38. Sergeev, A.N., Leites, D.: Casimir operators for Lie superalgebras. arXiv:0202180v1

Download references

Author information

Authors and Affiliations

  1. Mathematics Research Unit, FSTC, University of Luxembourg, 6, rue Coudenhove-Kalergi, 1359, Luxembourg, Luxembourg

    Pierre Mathonet

  2. Institute of Mathematics, University of Liège, Grande Traverse, 12-B37, 4000, Liège, Belgium

    Fabian Radoux

Authors
  1. Pierre Mathonet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabian Radoux
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabian Radoux.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mathonet, P., Radoux, F. Projectively Equivariant Quantizations over the Superspace \({\mathbb{R}^{p|q}}\) . Lett Math Phys 98, 311–331 (2011). https://doi.org/10.1007/s11005-011-0474-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-011-0474-0

Mathematics Subject Classification (2010)

Keywords

Search

Navigation