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

SIGMA 9 (2013), 055, 17 pages      arXiv:1302.3727      https://doi.org/10.3842/SIGMA.2013.055

### $\mathfrak{spo}(2|2)$-Equivariant Quantizations on the Supercircle $S^{1|2}$

Najla Mellouli a, Aboubacar Nibirantiza b and Fabian Radoux b
a) University of Sfax, Higher Institute of Biotechnology, Route de la Soukra km 4, B.P. no 1175, 3038 Sfax, Tunisia
b) University of Liège, Institute of Mathematics, Grande Traverse, 12 - B37, B-4000 Liège, Belgium

Received February 18, 2013, in final form August 15, 2013; Published online August 23, 2013

Abstract
We consider the space of differential operators $\mathcal{D}_{\lambda\mu}$ acting between $\lambda$- and $\mu$-densities defined on $S^{1|2}$ endowed with its standard contact structure. This contact structure allows one to define a filtration on $\mathcal{D}_{\lambda\mu}$ which is finer than the classical one, obtained by writting a differential operator in terms of the partial derivatives with respect to the different coordinates. The space $\mathcal{D}_{\lambda\mu}$ and the associated graded space of symbols $\mathcal{S}_{\delta}$ ($\delta=\mu-\lambda$) can be considered as $\mathfrak{spo}(2|2)$-modules, where $\mathfrak{spo}(2|2)$ is the Lie superalgebra of contact projective vector fields on $S^{1|2}$. We show in this paper that there is a unique isomorphism of $\mathfrak{spo}(2|2)$-modules between $\mathcal{S}_{\delta}$ and $\mathcal{D}_{\lambda\mu}$ that preserves the principal symbol (i.e. an $\mathfrak{spo}(2|2)$-equivariant quantization) for some values of $\delta$ called non-critical values. Moreover, we give an explicit formula for this isomorphism, extending in this way the results of [Mellouli N., SIGMA 5 (2009), 111, 11 pages] which were established for second-order differential operators. The method used here to build the $\mathfrak{spo}(2|2)$-equivariant quantization is the same as the one used in [Mathonet P., Radoux F., Lett. Math. Phys. 98 (2011), 311-331] to prove the existence of a $\mathfrak{pgl}(p+1|q)$-equivariant quantization on $\mathbb{R}^{p|q}$.

Key words: equivariant quantization; supergeometry; contact geometry; orthosymplectic Lie superalgebra.

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References

1. Berezin F.A., Introduction to superanalysis, Mathematical Physics and Applied Mathematics, Vol. 9, D. Reidel Publishing Co., Dordrecht, 1987.
2. Boniver F., Hansoul S., Mathonet P., Poncin N., Equivariant symbol calculus for differential operators acting on forms, Lett. Math. Phys. 62 (2002), 219-232, math.RT/0206213.
3. Boniver F., Mathonet P., IFFT-equivariant quantizations, J. Geom. Phys. 56 (2006), 712-730, math.RT/0109032.
4. Bouarroudj S., Projectively equivariant quantization map, Lett. Math. Phys. 51 (2000), 265-274, math.DG/0003054.
5. Cap A., Silhan J., Equivariant quantizations for AHS-structures, Adv. Math. 224 (2010), 1717-1734, arXiv:0904.3278.
6. Duval C., Lecomte P., Ovsienko V., Conformally equivariant quantization: existence and uniqueness, Ann. Inst. Fourier (Grenoble) 49 (1999), 1999-2029, math.DG/9902032.
7. Fox D.J.F., Projectively invariant star products, Int. Math. Res. Pap. (2005), 461-510, math.DG/0504596.
8. Gargoubi H., Mellouli N., Ovsienko V., Differential operators on supercircle: conformally equivariant quantization and symbol calculus, Lett. Math. Phys. 79 (2007), 51-65, math-ph/0610059.
9. Hansoul S., Projectively equivariant quantization for differential operators acting on forms, Lett. Math. Phys. 70 (2004), 141-153.
10. Hansoul S., Existence of natural and projectively equivariant quantizations, Adv. Math. 214 (2007), 832-864, math.DG/0601518.
11. Kac V.G., Lie superalgebras, Adv. Math. 26 (1977), 8-96.
12. Lecomte P.B.A., Classification projective des espaces d'opérateurs différentiels agissant sur les densités, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 287-290.
13. Lecomte P.B.A., Towards projectively equivariant quantization, Progr. Theoret. Phys. Suppl. (2001), no. 144, 125-132.
14. Lecomte P.B.A., Ovsienko V.Yu., Projectively equivariant symbol calculus, Lett. Math. Phys. 49 (1999), 173-196, math.DG/9809061.
15. Leites D., Poletaeva E., Serganova V., On Einstein equations on manifolds and supermanifolds, J. Nonlinear Math. Phys. 9 (2002), 394-425, math.DG/0306209.
16. Leuther T., Mathonet P., Radoux F., One ${\mathfrak{osp}}(p+1,q+1|2r)$-equivariant quantizations, J. Geom. Phys. 62 (2012), 87-99, arXiv:1107.1387.
17. Leuther T., Radoux F., Natural and projectively invariant quantizations on supermanifolds, SIGMA 7 (2011), 034, 12 pages, arXiv:1010.0516.
18. Mathonet P., Radoux F., Natural and projectively equivariant quantizations by means of Cartan connections, Lett. Math. Phys. 72 (2005), 183-196, math.DG/0606554.
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., On natural and conformally equivariant quantizations, J. Lond. Math. Soc. (2) 80 (2009), 256-272, arXiv:0707.1412.
21. 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.
22. Mathonet P., Radoux F., Projectively equivariant quantizations over the superspace ${\mathbb R}^{p|q}$, Lett. Math. Phys. 98 (2011), 311-331, arXiv:1003.3320.
23. Mellouli N., Second-order conformally equivariant quantization in dimension $1|2$, SIGMA 5 (2009), 111, 11 pages, arXiv:0912.5190.
24. Michel J.-P., Quantification conformément équivariante des fibrés supercotangents, Ph.D. thesis, Université de la Méditerranée - Aix-Marseille II, 2009, available at http://tel.archives-ouvertes.fr/tel-00425576.
25. Musson I.M., On the center of the enveloping algebra of a classical simple Lie superalgebra, J. Algebra 193 (1997), 75-101.
26. Pinczon G., The enveloping algebra of the Lie superalgebra ${\rm osp}(1,2)$, J. Algebra 132 (1990), 219-242.
27. Sergeev A., The invariant polynomials on simple Lie superalgebras, Represent. Theory 3 (1999), 250-280, math.RT/9810111.