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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn021, 19 pages, doi:10.1093/imrn/rnn021 published on May 1, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Supertransvectants and Symplectic Geometry

H. Gargoubi1 and V. Ovsienko2

1 Institut Préeparatoire Aux Etudes d'lngéenieurs, 2 Rue Jawaher Lel Nehru, Monfleury 1008 Tunis, Tunisia
2 Centre National de la Recherche Scientifique, Institut Camille Jordan, Universit ée Claude Bernard Lyon 1, 21 Avenue Claude Bernard, 69622 Villeurbanne Cedex, France

Correspondence: Correspondence to be sent to: hichem.gargoubi{at}ipeit.rnu.tn

The 1|1-supertransvectants are the osp(1|2)-invariant bilinear operations on weighted densities on the supercircle S1|1, the projective version of Formula . These operations are analogues of the famous Gordan transvectants (or Rankin–Cohen brackets). We prove that supertransvectants coincide with the iterated Poisson and ghost Poisson brackets on Formula and apply this result to construct star-products.



References

  1. Cohen H. Sums involving the values at negative integers of L-functions of quadratic characters. Mathematische Annalen (1975) 217:271–85.[CrossRef][ISI]
  2. Cohen P., Manin Yu., Zagier D. Automorphic Pseudodifferential Operators: Algebraic Aspects of Integrable Systems (1997) Boston, MA: Birkhäuser Boston. 17–47. Progress in Nonlinear Differential Equations Applications 26.
  3. Gargoubi H., Mellouli N., Ovsienko V. Differential operators on supercircle: conformally equivariant quantization and symbol calculus. Letters in Mathematical Physics (2007) 79:51–65.[CrossRef][ISI]
  4. El Gradechi A. M. The Lie theory of the Rankin-Cohen brackets and allied bi-differential operators. Advances in Mathematics (2006) 207:484–531.[CrossRef][ISI]
  5. Gieres F. Conformally covariant operators on Riemann surfaces (with application to conformal and integrable models). International Journal of Modern Physics A (1993) 8:1–58.[Medline]
  6. Gieres F., Theisen S. Superconformally covariant operators and super W-algebras. Journal of Mathematical Physics (1993) 34:5964–85.[CrossRef][ISI]
  7. Gordan P. Invariantentheorie (1887) Leipzig, Germany: Teubner.
  8. Grozman P., Leites D., Shchepochkina I. Lie superalgebras of string theories. Acta Mathematica Vietnamica (2001) 26(1):27–63.
  9. Huang W.-J. Superconformal covariantization of superdifferential operator on (1|1) superspace and classical N = 2W superalgebras. Journal of Mathematical Physics (1994) 35(5):2570–82.[CrossRef][ISI]
  10. Janson S., Peetre J. A new generalization of Hankel operators (the case of higher weights). Mathematische Nachrichten (1987) 132:313–28.[CrossRef][ISI]
  11. Kac V. G. Classification of Supersymmetries (2002) Beijing, China: Higher Education Press. 319–44. Proceedings of the International Congress of Mathematics 1.
  12. Kosmann-Schwarzbach Y. Derived brackets. Letters in Mathematical Physics (2004) 69:61–87.[CrossRef][ISI]
  13. Leites D. Lie Superalgebras (1984) Moscow, Russia: Akad. Sci. USSR. 3–49. Current Problems in Mathematics 25.
  14. Leites D., Kochetkov Yu., Weintrob A. New Invariant Differential Operators on Supermanifolds and Pseudo (co)homology (1991) New York: Dekker. 217–38. Lecture Notes in Pure and Applied Mathematics 134.
  15. Manin Yu. Topics in Non-commutative Geometry (1979) 1st ed. Princeton, NJ: Princeton University Press.
  16. Michel J.-P., Duval C. On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative. (2007) (preprint math-ph/0710.1544.
  17. Olver P. J., Sanders J. A. Transvectants, modular forms and the Heisenberg algebra. Advances in Applied Mathematics (2000) 25:252–83.[CrossRef][ISI]
  18. Omori H., Maeda Y., Miyazaki N., Yoshioka A. Deformation quantization of the Poisson algebra of Laurent polynomials. Letters in Mathematical Physics (1998) 46:171–80.[CrossRef][ISI]
  19. Ovsienko V. Exotic deformation quantization. Journal of Differential Geometry (1997) 45:390–406.[ISI]
  20. Ovsienko V., Tabachnikov S. Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups (2005) Cambridge, UK: Cambridge University Press. Cambridge Tracts in Mathematics 165.
  21. Rankin R. A. The construction of automorphic forms from the derivatives of a given form. Journal of the Indian Mathematical Society (1956) 20:103–16.
  22. Shchepochkina I. How to realize Lie algebras by vector fields. Theoretical and Mathematical Physics (2006) 147(3):821–38.[CrossRef][ISI]
  23. Zagier D. Modular forms and differential operators. Proceedings of the Indian Academy of Science (Mathematical Sciences) (1994) 104:57–75.

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This Article
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