Poisson cohomology of regular Poisson manifolds
Annales de l'Institut Fourier, Tome 42 (1992) no. 4, pp. 967-988.

Le but principal de cet article est de proposer une méthode pour calculer la cohomologie de Poisson d’une variété de Poisson par l’utilisation de groupoïdes symplectiques.

L’idée clé est de se ramener à calculer la cohomologie de Rham de certaines variétés. En particulier nous en déduisons une formule pour la cohomologie de Poisson d’une variété de Poisson régulière dont la feuilletage symplectique est trivial.

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.

@article{AIF_1992__42_4_967_0,
     author = {Xu, Ping},
     title = {Poisson cohomology of regular {Poisson} manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {967--988},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {42},
     number = {4},
     year = {1992},
     doi = {10.5802/aif.1317},
     mrnumber = {94d:58167},
     zbl = {0759.58020},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1317/}
}
TY  - JOUR
AU  - Xu, Ping
TI  - Poisson cohomology of regular Poisson manifolds
JO  - Annales de l'Institut Fourier
PY  - 1992
SP  - 967
EP  - 988
VL  - 42
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1317/
DO  - 10.5802/aif.1317
LA  - en
ID  - AIF_1992__42_4_967_0
ER  - 
%0 Journal Article
%A Xu, Ping
%T Poisson cohomology of regular Poisson manifolds
%J Annales de l'Institut Fourier
%D 1992
%P 967-988
%V 42
%N 4
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1317/
%R 10.5802/aif.1317
%G en
%F AIF_1992__42_4_967_0
Xu, Ping. Poisson cohomology of regular Poisson manifolds. Annales de l'Institut Fourier, Tome 42 (1992) no. 4, pp. 967-988. doi : 10.5802/aif.1317. http://www.numdam.org/articles/10.5802/aif.1317/

[BT] R. Bott, L.W. Tu, Differential forms in algebraic topology, Springer-Verlag, (1981).

[Br] J.-L. Brylinski, A differential complex for Poisson Manifolds, J. Diff. Geom., 28 (1988), 93-114. | MR | Zbl

[CDW] A. Coste, P. Dazord, A. Weinstein, Groupoïdes symplectiques, Publications du Départment de Mathématiques, Université Claude Bernard Lyon I, (1987). | Numdam | MR | Zbl

[D1] P. Dazord, Groupoïdes symplectiques et troisième théorème de Lie “non linéaire”, Lecture Notes in Mathematics, vol. 1416 (1990), 39-74. | MR | Zbl

[D2] P. Dazord, Réalisations isotropes de Libermann, Publ. Dept. Math. Lyon, (1989).

[DD] P. Dazord, and T. Delzant, le problème general des variables actions angles, J. Diff. Geom., 26 (1987), 223-251. | MR | Zbl

[F] D.B. Fuks, Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants Bureau (1986). | MR | Zbl

[Ka] M.V. Karasev, Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, Math. USSR Izvestiya, 28 (1987), 497-527. | Zbl

[H] J. Huebschmann, Poisson cohomology and quantization, J. reine angew. Math., 408 (1990), 57-113. | MR | Zbl

[L] A. Lichnerowicz, Les variétés de Poisson et leurs algebres de Lie associées, J. Diff. Geom., 12 (1977), 253-300. | MR | Zbl

[M] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry ; LMS lecture Notes Series, 124 Cambridge Univ. Press, (1987). | MR | Zbl

[V] I. Vaisman, Remarks on the Licherowicz-Poisson cohomology, Ann. Inst. Fourier, Grenoble, 40, 4 (1990), 951-963. | Numdam | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[W1] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557. | MR | Zbl

[W2] A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16 (1987), 101-104. | MR | Zbl

[WX] A. Weinstein, P. Xu, Extensions of symplectic groupoids and quantization, J. reine angew. Math., 417 (1991), 159-189. | MR | Zbl

Cité par Sources :