Existence of star-products on exact symplectic manifolds

Marc De Wilde; P. B. A. Lecomte

Annales de l'institut Fourier (1985)

  • Volume: 35, Issue: 2, page 117-143
  • ISSN: 0373-0956

Abstract

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It is shown that if a manifold admits an exact symplectic form, then its Poisson Lie algebra has non trivial formal deformations and the manifold admits star-products. The non-formal derivations of the star-products and the deformations of the Poisson Lie algebra of an arbitrary symplectic manifold are studied.

How to cite

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De Wilde, Marc, and Lecomte, P. B. A.. "Existence of star-products on exact symplectic manifolds." Annales de l'institut Fourier 35.2 (1985): 117-143. <http://eudml.org/doc/74672>.

@article{DeWilde1985,
abstract = {It is shown that if a manifold admits an exact symplectic form, then its Poisson Lie algebra has non trivial formal deformations and the manifold admits star-products. The non-formal derivations of the star-products and the deformations of the Poisson Lie algebra of an arbitrary symplectic manifold are studied.},
author = {De Wilde, Marc, Lecomte, P. B. A.},
journal = {Annales de l'institut Fourier},
keywords = {Chevalley cohomology; Hochschild cohomology; Poisson Lie algebra},
language = {eng},
number = {2},
pages = {117-143},
publisher = {Association des Annales de l'Institut Fourier},
title = {Existence of star-products on exact symplectic manifolds},
url = {http://eudml.org/doc/74672},
volume = {35},
year = {1985},
}

TY - JOUR
AU - De Wilde, Marc
AU - Lecomte, P. B. A.
TI - Existence of star-products on exact symplectic manifolds
JO - Annales de l'institut Fourier
PY - 1985
PB - Association des Annales de l'Institut Fourier
VL - 35
IS - 2
SP - 117
EP - 143
AB - It is shown that if a manifold admits an exact symplectic form, then its Poisson Lie algebra has non trivial formal deformations and the manifold admits star-products. The non-formal derivations of the star-products and the deformations of the Poisson Lie algebra of an arbitrary symplectic manifold are studied.
LA - eng
KW - Chevalley cohomology; Hochschild cohomology; Poisson Lie algebra
UR - http://eudml.org/doc/74672
ER -

References

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  1. [1] A. AVEZ, A. LICHNEROWICZA. DIAZ-MIRANDA, Sur l'algèbre de Lie des automorphismes infinitésimaux d'une variété symplectique, J. Diff. Geom., 9 (1974), 1-40. Zbl0283.53033MR50 #8602
  2. [2] F. BAYEN, M. FLATO, C. FRONSDAL, A. LICHNEROWICZ, D. STERNHEIMER, Deformation theory and quantization, Ann. of Physics, 111 (1978), 61-110 and 111-151. Zbl0377.53024MR58 #14737a
  3. [3] M. CAHEN, S. GUTT, Regular star-representation of Lie algebras, Lett. in Math. Physics, 6 (1982), 395-404. Zbl0522.58018MR84d:58025
  4. [4] M. DE WILDE, S. GUTT, P. LECOMTE, A propos des deuxième et troisième espaces de cohomologie de l'algèbre de Lie de Poisson d'une variété symplectique, Ann. Inst. Poincaré, 40, 1 (1984), 77-93. Zbl0547.53024MR86a:58036
  5. [5] M. DE WILDE, P. LECOMTE, Cohomology of the Lie algebra of smooth vector fields of a manifold, associated to the Lie derivative of smooth forms, J. Math. Pures and Appl., 62 (1983), 197-214. Zbl0481.58032MR85j:17017
  6. [6] M. DE WILDE, P. LECOMTE, Star-products on cotangent bundles, Lett. in Math. Phys., 7 (1983), 487-496. Zbl0526.58023
  7. [7] M. FLATO, A. LICHNEROWICZ, D. STERNHEIMER, C.R.A.S., Paris, 279 (1974), 877. Zbl0289.53031
  8. [8] S. GUTT, Second et troisième espaces de cohomologie différentiable de l'algèbre de Lie de Poisson d'une variété symplectique, Ann. Inst. Poincaré, 33-1 (1981), 1-31. Zbl0476.53021
  9. [9] A. LICHNEROWICZ, Déformations d'algèbres associées à une variété symplectique (les *v-produits), Ann. Inst. Fourier, 32-1 (1982), 157-209. Zbl0465.53025MR83k:58095
  10. [10] A. LICHNEROWICZ, Sur les algèbres formelles associées par déformation à une variété symplectique, Ann. Di Math., 123 (1980), 287-330. Zbl0441.53029MR82m:58030
  11. [11] O. M. NEROSLAVSKY, A. T. VLASSOV, Sur les déformations de l'algèbre des fonctions d'une variété symplectique, C.R.A.S., Paris, (1980). Zbl0471.58034
  12. [12] A. NUYENHUIS, R. RICHARDSON, Deformation of Lie algebra structures, J. of Math. and Mechanics, 17-1 (1967), 89-105. Zbl0166.30202
  13. [13] J. VEY, Déformation du crochet de Poisson d'une variété symplectique, Comm. Math. Helv., 50 (1975), 421-454. Zbl0351.53029MR54 #8765

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