Poisson cohomology of regular Poisson manifolds

Ping Xu

Annales de l'institut Fourier (1992)

  • Volume: 42, Issue: 4, page 967-988
  • ISSN: 0373-0956

Abstract

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The main purpose of this paper is to suggest a method of computing Poisson cohomology of a Poisson manifold by means of symplectic groupoids. The key idea is to convert the problem of computing Poisson cohomology to that of computing de Rham cohomology of certain manifolds. In particular, we shall derive an explicit formula for the Poisson cohomology of a regular Poisson manifold where the symplectic foliation is a trivial fibration.

How to cite

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Xu, Ping. "Poisson cohomology of regular Poisson manifolds." Annales de l'institut Fourier 42.4 (1992): 967-988. <http://eudml.org/doc/74981>.

@article{Xu1992,
abstract = {The main purpose of this paper is to suggest a method of computing Poisson cohomology of a Poisson manifold by means of symplectic groupoids. The key idea is to convert the problem of computing Poisson cohomology to that of computing de Rham cohomology of certain manifolds. In particular, we shall derive an explicit formula for the Poisson cohomology of a regular Poisson manifold where the symplectic foliation is a trivial fibration.},
author = {Xu, Ping},
journal = {Annales de l'institut Fourier},
keywords = {Poisson cohomology; symplectic groupoids; de Rham cohomology},
language = {eng},
number = {4},
pages = {967-988},
publisher = {Association des Annales de l'Institut Fourier},
title = {Poisson cohomology of regular Poisson manifolds},
url = {http://eudml.org/doc/74981},
volume = {42},
year = {1992},
}

TY - JOUR
AU - Xu, Ping
TI - Poisson cohomology of regular Poisson manifolds
JO - Annales de l'institut Fourier
PY - 1992
PB - Association des Annales de l'Institut Fourier
VL - 42
IS - 4
SP - 967
EP - 988
AB - The main purpose of this paper is to suggest a method of computing Poisson cohomology of a Poisson manifold by means of symplectic groupoids. The key idea is to convert the problem of computing Poisson cohomology to that of computing de Rham cohomology of certain manifolds. In particular, we shall derive an explicit formula for the Poisson cohomology of a regular Poisson manifold where the symplectic foliation is a trivial fibration.
LA - eng
KW - Poisson cohomology; symplectic groupoids; de Rham cohomology
UR - http://eudml.org/doc/74981
ER -

References

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  2. [Br] J.-L. BRYLINSKI, A differential complex for Poisson Manifolds, J. Diff. Geom., 28 (1988), 93-114. Zbl0634.58029MR89m:58006
  3. [CDW] A. COSTE, P. DAZORD, A. WEINSTEIN, Groupoïdes symplectiques, Publications du Départment de Mathématiques, Université Claude Bernard Lyon I, (1987). Zbl0668.58017MR90g:58033
  4. [D1] P. DAZORD, Groupoïdes symplectiques et troisième théorème de Lie “non linéaire”, Lecture Notes in Mathematics, vol. 1416 (1990), 39-74. Zbl0702.58023MR91i:58169
  5. [D2] P. DAZORD, Réalisations isotropes de Libermann, Publ. Dept. Math. Lyon, (1989). 
  6. [DD] P. DAZORD, and T. DELZANT, le problème general des variables actions angles, J. Diff. Geom., 26 (1987), 223-251. Zbl0634.58003MR88j:58032
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  8. [Ka] M.V. KARASEV, Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, Math. USSR Izvestiya, 28 (1987), 497-527. Zbl0624.58007
  9. [H] J. HUEBSCHMANN, Poisson cohomology and quantization, J. reine angew. Math., 408 (1990), 57-113. Zbl0699.53037MR92e:17027
  10. [L] A. LICHNEROWICZ, Les variétés de Poisson et leurs algebres de Lie associées, J. Diff. Geom., 12 (1977), 253-300. Zbl0405.53024MR58 #18565
  11. [M] K. MACKENZIE, Lie groupoids and Lie algebroids in differential geometry ; LMS lecture Notes Series, 124 Cambridge Univ. Press, (1987). Zbl0683.53029MR89g:58225
  12. [V] I. VAISMAN, Remarks on the Licherowicz-Poisson cohomology, Ann. Inst. Fourier, Grenoble, 40, 4 (1990), 951-963. Zbl0708.58010MR92c:58155
  13. [VK1] Yu. M. VOROB'EV, M.V. KARASEV, Corrections to classical dynamics and quantization conditions which arise in the deformation of Poisson brackets, Dokl. Akad. Nauk USSR, 247, No. 6 (1987), 1294-1298. Zbl0676.58026
  14. [VK2] Yu. M. VOROB'EV, M.V. KARASEV, Poisson manifolds and the Schouten Bracket, Functional Analysis and its Applications, Vol. 22, No. 1 (1988), 1-9. Zbl0667.58018MR89k:58011
  15. [W1] A. WEINSTEIN, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557. Zbl0524.58011MR86i:58059
  16. [W2] A. WEINSTEIN, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16 (1987), 101-104. Zbl0618.58020MR88c:58019
  17. [WX] A. WEINSTEIN, P. XU, Extensions of symplectic groupoids and quantization, J. reine angew. Math., 417 (1991), 159-189. Zbl0722.58021MR92k:58094

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