Mathematical Notes

, Volume 92, Issue 3–4, pp 302–311 | Cite as

Cohomology of osp(2|2) acting on spaces of linear differential operators on the superspace ℝ1|2

  • N. Ben FrajEmail author
  • M. Boujelben


We compute the first differential cohomology of the orthosymplectic Lie superalgebra osp(2|2) with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1, 2)-dimensional real superspace. We also compute the same, but osp(1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomologies. This work is the simplest generalization of a result from [1].


differential cohomology orthosymplectic Lie superalgebra differential operator superspace Lie algebra of contact vector fields 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. Basdouri and M. Ben Ammar, “Cohomology of osp(1|2) acting on linear differential operators on the supercircle S 1|1,” Lett. Math. Phys. 81(3), 239–251 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    P. B. A. Lecomte, “On the cohomology of sl(m + 1;ℝ) acting on differential operators and sl(m + 1;ℝ)-equivariant symbols,” Indag. Math. (N. S.) 11(1), 95–114 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    B. Agrebaoui, F. Ammar, P. Lecomte, and V. Ovsienko, “Multi-parameter deformations of the module of symbols of differential operators,” Int. Math. Res. Not. 16, 847–869 (2002).MathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Agrebaoui, M. Ben Ammar, N. Ben Fraj, and V. Ovsienko, “Deformations of modules of differential forms,” J. Nonlinear Math. Phys. 10(2), 148–156 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    M. Ben Ammar and M. Boujelbene, “sl(2)-Trivial Deformations of VectPol(ℝ)-Modules of Symbols,” SIGMA 4 (065) (2008).Google Scholar
  6. 6.
    A. Nijenhuis and R. W. Richardson, Jr., “Deformations of homomorphisms of Lie groups and Lie algebras,” Bull. Amer. Math. Soc. 73, 175–179 (1967).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    H. Gargoubi, N. Mellouli, and V. Ovsienko, “Differential operators on supercircle: Conformally equivariant quantization and symbol calculus,” Lett. Math. Phys. 79(1), 51–65 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    B. L. Feigin and D. B. Fucks, “Cohomology of Lie groups and Lie algebras,” in Lie Groups and Lie Algebras II, Current Problems in Mathematics. Fundamental Directions, Itogi Nauki i Tekhniki [Progress in Science and Technology] (VINITI, Moscow, 1988), Vol. 21, pp. 121–209 [Encycl. Math. Sci. 21, 125–215 (2000)].Google Scholar
  9. 9.
    P. Grozman, D. Leites, and I. Shchepochkina, “Lie superalgebras of string theories,” Acta Math. Vietnam 26(1), 27–63 (2001); arXiv: hep-th/9702120MathSciNetzbMATHGoogle Scholar
  10. 10.
    C. H. Conley, “Conformal symbols and the action of contact vector fields over the superline,” J. Reine Angew. Math. 633, 115–163 (2009); arXiv: math. RT/0712.1780v2 (2008).MathSciNetzbMATHGoogle Scholar
  11. 11.
    N. Ben Fraj, “Cohomology of K(2) acting on linear differential operators on the superspace ℝ1|2,” Lett. Math. Phys. 86(2–3), 159–175 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    I. Basdouri, M. Ben Ammar, N. Ben Fraj, M. Boujelben, and K. Kammoun, “Cohomology of the Lie superalgebra of contact vector fields on K1|1 and deformations of the superspace of symbols,” J. Nonlinear Math. Phys. 16(4), 373–409 (2009).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Institut Supérieur de Sciences Appliquées et TechnologieSousseTunisie

Personalised recommendations