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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
- 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.
- 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.
- 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.
- 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.
- Fialowski A., Deformations of Lie algebras, Mat. Sb.
55 (1986), 467-473.
- 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.
- Fialowski A., Fuchs D.B., Construction of miniversal
deformations of Lie algebras, J. Funct. Anal. 161
(1999), 76-110,
math.RT/0006117.
- Fuchs D.B., Cohomology of infinite-dimensional Lie algebras,
Consultants Bureau, New York, 1987.
- Gargoubi H., Sur la géométrie de l'espace des
opérateurs différentiels linéaires sur R,
Bull. Soc. Roy. Sci. Liège 69 (2000), 21-47.
- 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.
- Gordan P., Invariantentheorie, Teubner, Leipzig, 1887.
- Nijenuis A., Richardson R.W. Jr., Deformations of
homomorphisms of Lie groups and Lie algebras, Bull. Amer.
Math. Soc. 73 (1967), 175-179.
- 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.
- 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.
- Richardson R.W., Deformations of subalgebras of Lie algebras,
J. Differential Geom. 3 (1969), 289-308.
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