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

SIGMA 4 (2008), 065, 19 pages      math.RT/0702712      https://doi.org/10.3842/SIGMA.2008.065
Contribution to the Special Issue on Deformation Quantization

sl(2)-Trivial Deformations of VectPol(R)-Modules of Symbols

Mabrouk Ben Ammar and Maha Boujelbene
Département de Mathématiques, Faculté des Sciences de Sfax, BP 802, 3038 Sfax, Tunisie

Received January 14, 2008, in final form September 05, 2008; Published online September 18, 2008
Mistake in Proposition 4 and further computations have been corrected November 18, 2008.

Abstract
We consider the action of VectPol(R) by Lie derivative on the spaces of symbols of differential operators. We study the deformations of this action that become trivial once restricted to sl(2). Necessary and sufficient conditions for integrability of infinitesimal deformations are given.

Key words: tensor densities, cohomology, deformations.

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References

  1. Agrebaoui B., Ben Fraj N., Ben Ammar M., Ovsienko V., Deformation of modules of differential forms, J. Nonlinear Math. Phys. 10 (2003), 148-156, math.QA/0310494.
  2. Agrebaoui B., Ammar F., Lecomte P., Ovsienko V., Multi-parameter deformations of the module of symbols of differential operators, Int. Math. Res. Not. 2002 (2002), no. 16, 847-869, math.QA/0011048.
  3. Bouarroudj S., On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys. 14 (2007), 112-127, math.DG/0502372.
  4. Bouarroudj S., Ovsienko V., Three cocycles on Diff(S1) generalizing the Schwarzian derivative, Int. Math. Res. Not. 1998 (1998), no. 1, 25-39, dg-ga/9710018.
  5. Fialowski A., Deformations of Lie algebras, Mat. Sb. 55 (1986), 467-473.
  6. Fialowski A., An example of formal deformations of Lie algebras, in Deformation Theory of Algebras and Structures and Applications, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 247, Kluwer Acad. Publ., Dordrecht, 1988, 375-401.
  7. Fialowski A., Fuchs D.B., Construction of miniversal deformations of Lie algebras, J. Funct. Anal. 161 (1999), 76-110, math.RT/0006117.
  8. Fuchs D.B., Cohomology of infinite-dimensional Lie algebras, Consultants Bureau, New York, 1987.
  9. Gargoubi H., Sur la géométrie de l'espace des opérateurs différentiels linéaires sur R, Bull. Soc. Roy. Sci. Liège 69 (2000), 21-47.
  10. 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.
  11. Gordan P., Invariantentheorie, Teubner, Leipzig, 1887.
  12. Nijenuis A., Richardson R.W. Jr., Deformations of homomorphisms of Lie groups and Lie algebras, Bull. Amer. Math. Soc. 73 (1967), 175-179.
  13. Ovsienko V., Roger C., Deforming the Lie algebra of vector fields on S1 inside the Lie algebra of pseudodifferential operators on S1, in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, Vol. 194, Amer. Math. Soc., Providence, RI, 1999, 211-226, math.QA/9812074.
  14. Ovsienko V., Roger C., Deforming the Lie algebra of vector fields on S1 inside the Poisson algebra on T*S1, Comm. Math. Phys. 198 (1998), 97-110, q-alg/9707007.
  15. Richardson R.W., Deformations of subalgebras of Lie algebras, J. Differential Geom. 3 (1969), 289-308.

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