Abstract
We consider the \(\mathfrak{sl}\, (2)\)-module structure on the spaces of symbols of differential operators acting on the spaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this structure and prove that any formal deformation is equivalent to its infinitesimal part. We study also the super analogue of this problem getting the same results.
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B. Agrebaoui, F. Ammar, P. Lecomte and V. Ovsienko, Multi-parameter deformations of the module of symbols of differential operators, Internat. Mathem. Research Notices, 16 (2002), 847–869.
B. Agrebaoui, N. Ben Fraj, M. Ben Ammar and V. Ovsienko, Deformation of modules of differential forms, Nonlinear Math. Physics, 10 (2003), 148–156.
I. Basdouri and M. Ben Ammar, Cohomology of \(\mathop {\mathfrak {osp}}\nolimits \, (1|2)\) acting on linear differential operators on the supercercle S 1|1, Letters in Mathematical Physics, 81 (2007), 239–251.
I. Basdouri, M. Ben Ammar, N. Ben Fraj, M. Boujelbene and K. Kammoun, Cohomology of the Lie superalgebra of contact vector fields on ℝ1|1 and deformations of the superspace of symbols, J. Nonlinear Math. Physics, 16 (2009), 1–37.
I. Basdouri, M. Ben Ammar, B. Dali and S. Omri, Deformation of \(\mathop {\mathfrak {vect}}\nolimits \, (1)\) -Modules of Symbols, doi:10.1016/j.geomphys.2009.12.002.
M. Ben Ammar and M. Boujelbene, \(\mathop {\mathfrak {sl}}\nolimits \, (2)\)-trivial deformation of \(\mathrm{Vect_{Pol}}(\mathbb{R})\)-modules of symbols, SIGMA, 4 (2008), 065. 19 pages
F. Berezin, Introduction to Superanalysis, Mathematical Physics and Applied Mathematics 9, Reidel (Dordrecht, 1987).
B. L. Feigin and D. B. Fuchs, Homology of the Lie algebras of vector fields on the line, Func. Anal. Appl., 14 (1980), 201–212.
A. Fialowski, Deformations of Lie algebras, Mat. Sbornik, USSR, 127(169) (1985), 476–482; English translation, Math. USSR-Sb., 55 (1986), 467–473.
A. Fialowski, An example of formal deformations of Lie algebras, in: Proceedings of the NATO Conference on Deformation Theory of Algebras and Applications (Il Ciocco, Italy, 1986), Kluwer (Dordrecht, 1988), pp. 375–401.
H. Gargoubi, N. Mellouli and V. Ovsienko, Differential operators on supercircle: Conformally equivariant quantization and symbol calculus, Letters in Mathematical Physics, 79 (2007), 51–65.
P. B. A. Lecomte, On the cohomology of \(\mathop {\mathfrak {sl}}\nolimits \, (n + 1;\mathbb{R})\) acting on differential operators and \(\mathop {\mathfrak {sl}}\nolimits \, (n + 1;\mathbb{R})\)-equivariant symbols, Indag. Math. NS., 11 (2000), 95–114.
D. Leites, Supermanifold Theory, USSR Academy of Science Petrozavodsk (1983) (in Russian).
A. Nijenuis and R. W. Richardson Jr., Deformations of homomorphisms of Lie groups and Lie algebras, Bull. Amer. Math. Soc., 73 (1967), 175–179.
V. Ovsienko and C. Roger, Deforming the Lie algebra of vector fields on S 1 inside the Lie algebra of pseudodifferential operators on S 1, AMS Transl. Ser. 2, (Adv. Math. Sci.), 194 (1999), 211–227.
V. Ovsienko and C. Roger, Deforming the Lie algebra of vector fields on S 1 inside the Poisson algebra on \(\dot{T}^{*}S^{1}\), Comm. Math. Phys., 198 (1998), 97–110.
R. W. Richardson, Deformations of subalgebras of Lie algebras, J. Diff. Geom., 3 (1969), 289–308.
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Basdouri, I., Ben Ammar, M. Deformation of \(\mathfrak{sl}\, (2)\) and \(\mathfrak{osp}\, (1|2)\)-modules of symbols. Acta Math Hung 137, 214–223 (2012). https://doi.org/10.1007/s10474-012-0220-9
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DOI: https://doi.org/10.1007/s10474-012-0220-9