Tohoku Mathematical Journal

Modules of bilinear differential operators over the orthosymplectic superalgebra $\mathfrak{osp}(1|2)$

Taher Bichr, Jamel Boujelben, and Khaled Tounsi

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Abstract

Let $\frak{F}_\lambda, \lambda\in \mathbb{C}$, be the space of tensor densities of degree $\lambda$ on the supercircle $S^{1|1}$. We consider the superspace $\mathfrak{D}_{\lambda_1,\lambda_2,\mu}$ of bilinear differential operators from $\frak{F}_{\lambda_1}\otimes\frak{F}_{\lambda_2}$ to $\frak{F}_{\mu}$ as a module over the orthosymplectic superalgebra $\mathfrak{osp}(1|2)$. We prove the existence and the uniqueness of a canonical conformally equivariant symbol map from $\mathfrak{D}_{\lambda_1,\lambda_2,\mu}^k$ to the corresponding space of symbols. An explicit expression of the associated quantization map is also given.

Article information

Source
Tohoku Math. J. (2), Volume 70, Number 2 (2018), 319-338.

Dates
First available in Project Euclid: 2 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1527904824

Digital Object Identifier
doi:10.2748/tmj/1527904824

Mathematical Reviews number (MathSciNet)
MR3810243

Zentralblatt MATH identifier
06929337

Subjects
Primary: 53D10: Contact manifolds, general
Secondary: 17B66: Lie algebras of vector fields and related (super) algebras 17B10: Representations, algebraic theory (weights)

Keywords
Bilinear differential operators densities orthosymplectic algebra symbol and quantization maps

Citation

Bichr, Taher; Boujelben, Jamel; Tounsi, Khaled. Modules of bilinear differential operators over the orthosymplectic superalgebra $\mathfrak{osp}(1|2)$. Tohoku Math. J. (2) 70 (2018), no. 2, 319--338. doi:10.2748/tmj/1527904824. https://projecteuclid.org/euclid.tmj/1527904824


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References

  • N. Belghith, M. Ben Ammar and N. Ben Fraj, Differential Operators on the Weighted Densities on the Supercircle $S^{1|n}$, arXiv:1306.0101v3 [math.DG].
  • J. Boujelben, T. Bichr and K. Tounsi, Bilinear differential operators: Projectively equivariant symbol and quantization maps, Tohoku Math. J. 67 (2015), no. 4, 481–493.
  • B. Berezin, Introduction to superanalysis, Math. Phys. Appl. Math, D. Reidel Publishing Co., Dordrecht, 1987.
  • S. Bouarroudj, Projectively equivariant quantization map, Lett. Math. Phys, 51 (2000), 265–274, math.DG/0003054.
  • A. Čap and J. Šilhan, Equivariant quantizations for AHS-structures, Adv. Math, 224 (2010), 1717–1734, arXiv:0904.3278.
  • C. Duval, P. Lecomte and V. Ovsienko, Conformally equivariant quantization: existence and uniqueness, Ann. Inst. Fourier. 49 (1999), no. 6, 1999–2029.
  • H. Gargoubi, Sur la géométrie de l'espace des opérateurs différentiels linéaires sur $\mathbb{R}$, Bull. Soc. Roy. Sci. Liège 69 (2000), no. 1, 21–47.
  • H. Gargoubi and V. Ovsienko, Modules of differential operators on the real line, Funct. Anal. Appl. 35 (2001), no. 1, 13–18.
  • D. J. F. Fox, Projectively invariant star products, Int. Math. Res. Pap. (2005), 461–510, math.DG/0504596.
  • H. Gargoubi, N. Mallouli and V. Ovsienko, Differential operators on supercircle: Conformally equivariant quantization and symbol calculus, Lett. Math. Phys. 79 (2007), no. 1, 51–65.
  • S. Hansoul, Projectively equivariant quantization for differential operators acting on forms, Lett. Math. Phys. 70 (2004), 141–153.
  • S. Hansoul, Existence of natural and projectively equivariant quantizations, Adv. Math. 214 (2007), 832–864, math.DG/0601518.
  • P. B. A. Lecomte, Classification projective des espaces d'opérateurs différentiels agissant sur les densités, C. R. Acad. Sci. Paris. Sér. I Math. 328 (1999), no. 4, 287–290.
  • P. B. A. Lecomte, Towards projectively equivariant quantization, Progr. Theoret. Phys. Suppl. No. 144 (2001), 125–132.
  • P. Lecomte and V. Ovsienko, Projectively equivariant symbol calculus, Lett. Math. Phys. 49 (1999), no. 3, 173–196.
  • T. Leuther, P. Mathonet and F. Radoux, One $osp(p+1, q+1|2r)$-equivariant quantizations, J. Geom. Phys. 62 (2012), no. 1, 87–99, arXiv:1107.1387.
  • T. Leuther and F. Radoux, Natural and projectively invariant quantizations on supermanifolds, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), 034, 12 pages, arXiv:1010.0516.
  • P. Mathonet and F. Radoux, Natural and projectively equivariant quantizations by means of Cartan connections, Lett. Math. Phys. 72 (2005), no. 3, 183–196, math.DG/0606554.
  • P. Mathonet and F. Radoux, Cartan connections and natural and projectively equivariant quantizations, J. Lond. Math. Soc. (2) 76 (2007), no. 1, 87–104, math.DG/0606556.
  • P. Mathonet and F. Radoux, On natural and conformally equivariant quantizations, J. Lond. Math. Soc. (2) 80 (2009), no. 1, 256–272, arXiv:0707.1412.
  • P. Mathonet and F. Radoux, Existence of natural and conformally invariant quantizations of arbitrary symbols, J. Nonlinear. Math. Phys. 17 (2010), 539–556, arXiv:0811.3710.
  • P. Mathonet and F. Radoux, Projectively equivariant quantizations over the superspace $\mathbb{R}^{p|q}$, Lett. Math. Phys. 98 (2011), 311–331, arXiv:1003.3320.
  • N. Mellouli, Second-order conformally equivariant quantization in dimension $1|2$, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 111, 11 pages, arXiv:0912.5190.
  • N. Mellouli, A. Nibirantiza and F. Radoux, $spo(2|2)$-Equivariant Quantizations on the Supercircle $S^{1|2}$, SIGMA Symmetry Integrability Geom. Methods Appl. 9 (2013), Paper 055, 17 pages.
  • I. Shchepochkina, How to realize a Lie algebra by vector fields, Theoret. and Math. Phys. 147 (2006), no. 3, 821–838, math.RT/0509472.