Nontrivial Deformation of Vec(R)-Modules of Symbols
Keywords:
nonrelative Cohomology, affine Lie subalgebra, nontrivial Deformation,Abstract
The action of Lie algebra of vector fields on the space of symbols is given by the Lie derivative. We compute the nonrelative cohomology space with respact the Lie subalgebra affine. We study nontrivial deformations of this action.
Downloads
References
[1] M. Afi, O. Basdouri, On the isomorphism of non-abelian extensions of n-Lie algebras, J. Geom. Phys. Volume 173, March 2022, 104427. 1
[2] B. Agrebaoui, F. Ammar, B.P. Lecomte On the Cohomologie of the space of differentials operators acting on skew-symmetric tensor fields or on forms, as moduls of the Lie algebrea of vector fields. Differential Geometry and its Applications 20 (2004). 241–249.
[3] B. Agrebaoui I. Basdouri, S. Hammami, S. Saidi, Deforming the orthosymplectic Lie superalgebra osp(3|2) inside the Lie superalgebra of superpseudodifferential operators SψDO(3). Accept in Advanced Studies Euro-Tbilisi Mathematical Journal. 3.2
[4] I. Basdouri, S. Benabdelhafidh, A. Saghrouni, S. Silvestrov, Cohomology and deformations of crossed homomorphisms on Lie conformal algebras. October 2025 Advanced Studies Euro-Tbilisi Mathematical Journal 18(3). 3.2
[5] I. Basdouri, M. Ben Ammar, B. Dali and S. Omri, Deformation of VectP(R)-Modules of Symbols. math. RT/0702664 (2007). J. Geom. Phys. 60 (2010), no. 3, 531-542. MR 2600013. 1
[6] I. Basdouri, M. Ben Ammar, Cohomology of osp(1|2) Acting on Linear Differential Operators on the Supercercle S1|1. Letters in Mathematical Physics(2007)81:239–251.
[7] I. Basdouri, M. Ben Ammar, Deformations of sl(2) and osp(1|2)-modules of symbols. Acta Mathematica Hungarica, Volume 137, Issue 3, Year 2012. 1
[8] 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 . Jour of Nonlinear Math Physics, Vol. 16, No. 4 (2009) 1–37. 1
[9] I. Basdouri, M. Ben Nasr, S. Chouaibi and H. Mechi, Construction of supermodular forms using differential operators from a given supermodular form. Journal of Geometry and Physics, August 2019. DOI: 10.1016/j.geomphys.2019.103488. 1
[10] I. Basdouri, D. Ammar, S. Saidi, Second Cohomology of aff(1) Acting on n-ary Differential Operators. Bull. Korean Math. Soc. 56(1): 13-22 January 2019. 3.2
[11] K. Basdouri et al. The Binary aff(n|1)-Invariant Differential Operators On Weighted Densities On The Superspace R1|n And aff(n|1)-Relative Cohomology. Journal of Geometry and Physics, Volume 123, p. 235-245. 2.1, 2.4
[12] K. Basdouri et al. Cohomology and Deformation of aff(1|1) Acting on Differential Operators. International Journal of Geometric Methods in Modern Physics, Dec 2017. 1
[13] K. Basdouri, G.Chaabane. Cohomology of Bihom-Lie superbialgebras. Journal of Geometry and Physics, Volume 221, March 2026, 105714. 1
[14] M. Ben Ammar, M. Boujelbene, sl(2)−Trivial Deformation of VectP(R)-Modules of Symbols. math. RT/0702712 (2007). SIGMA 4 (2008), 065, 19 pages. 1, 3.2, 3.3, 3.3
[15] S. Bouarroudj, On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys., no.1, (2007), 112-127.
[16] S. Chouaibi, A. Zbidi, Rankin Cohen brackets on supermodular forms. Journal of Geometry and Physics, August 2019.
[17] B. L. Feigin D.B. Fuks, Homology of the Lie algebras of vector fields on the line, Func. Anal. Appl., 14 (1980) 201- 212. 1
[18] A. Fialowski, D. B. Fuchs, Construction of miniversal deformations of Lie algebras J. Func. Anal. 161:1 (1999) 76–110. 2.1, 2.4
[19] D. B. Fuchs, Cohomology of infinite-dimensional Lie algebras, Consultants Bureau, New York, 1987. 1, 3.2, 3.3, 3.3
[20] P. Gupta1, P. Dhar, Devranjan Samanta1 Rheology and electro-magnetism stimulated non-trivial deformation dynamics of viscoelastic compound droplets. Proceedings of the Royal Society A, May 2025 Volume 481Issue 2314.
[21] P. Lecomte, V. Ovsienko, Cohomology of the vector fields Lie algebra and modules of differential operators on a smooth manifold, Compositio Mathematica 124:1 (2000) 95–110. 1, 3.3
[22] A. Nijenuis, R. W. Richardson Jr., Deformations of homomorphisms of Lie groups and Lie algebras, Bull. Amer. Math. Soc. 73 (1967), 175–179.
[23] V. Ovsienko, C. Roger, Deforming the Lie algebra of vector fields on S1 inside the Lie algebra of pseudodifferential operators on S1, AMS Transl. Ser. 2, (Adv. Math. Sci.) vol. 194 (1999) 211–227.
[24] V. Ovsienko, C. Roger, Deforming the Lie algebra of vector fields on S1 inside the Poisson algebra on ˙T∗S1, Comm. Math. Phys., 198 (1998) 97–110.
[25] E.V. Vtorushin, V.N. Dorovsky Application of nonstationary nonEuclidean model of inelastic deformations to rock cutting. Journal of Petroleum Science and Engineering Volume 177, June 2019, Pages 508-517
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Khaled Basdouri
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
