Geodesic
The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces
Canadian mathematical bulletin, Tome 58 (2015) no. 3, pp. 575-579

Voir la notice de l'article provenant de la source Cambridge University Press

A surface $\sum$ endowed with a Poisson tensor $\pi$ is known to admit a canonical integration, $G\left( \pi\right)$ , which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if $\text{ }\!\!\pi\!\!\text{ }$ is not an area form on the 2-sphere, then $G\left( \pi\right)$ is diffeomorphic to the cotangent bundle $T*\sum$ . This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.
DOI : 10.4153/CMB-2015-011-7
Mots-clés : 58H05, 55R10, 53D17, Poisson tensor, Lie groupoid, cotangent bundle 
Torres, David Martínez. The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces. Canadian mathematical bulletin, Tome 58 (2015) no. 3, pp. 575-579. doi: 10.4153/CMB-2015-011-7
@article{10_4153_CMB_2015_011_7,
     author = {Torres, David Mart{\'\i}nez},
     title = {The {Diffeomorphism} {Type} of {Canonical} {Integrations} of {Poisson} {Tensors} on {Surfaces}},
     journal = {Canadian mathematical bulletin},
     pages = {575--579},
     year = {2015},
     volume = {58},
     number = {3},
     doi = {10.4153/CMB-2015-011-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2015-011-7/}
}
TY  - JOUR
AU  - Torres, David Martínez
TI  - The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces
JO  - Canadian mathematical bulletin
PY  - 2015
SP  - 575
EP  - 579
VL  - 58
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2015-011-7/
DO  - 10.4153/CMB-2015-011-7
ID  - 10_4153_CMB_2015_011_7
ER  - 
%0 Journal Article
%A Torres, David Martínez
%T The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces
%J Canadian mathematical bulletin
%D 2015
%P 575-579
%V 58
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2015-011-7/
%R 10.4153/CMB-2015-011-7
%F 10_4153_CMB_2015_011_7

[1] [1] Cattaneo, A. S. and Felder, G., Poisson sigma models and symplecticgroupoids.In: Quantization of singularsymplectic quotients, Progr. Math., 198, Birkhâuser, Basel, 2001, pp. 61–93. Google Scholar

[2] [2] Bonechi, F., Ciccoli, N., Staffolani, N., and Tarlini, M., The quantization of the symplecticgroupoid of the standard Podles sphere. J. Geom. Phys. 62(2012), no. 8,1851–1865. http://dx.doi.Org/1 0.101 6/j.geomphys.2012.04.001 Google Scholar

[3] [3] Crainic, M. and Fernandes, R. L., Integrability of Poisson brackets. J. Differential Geom. 66(2004), no. 1, 71–137. Google Scholar

[4] [4] Crainic, M. and Màrcuf, I., On the existence of symplectic realizations. J. Symplectic Geom. 9(2011), no. 4, 435–444. Google Scholar | DOI

[5] [5] Hawkins, E., A groupoid approach to quantization. J. Symplectic Geom. 6(2008), no. 1, 61–125. Google Scholar | DOI

[6] [6] Martinez Torres, D., A note on the separability of canonical integrations of Lie algebroids. Math. Res. Lett. 17(2010), no. 1, 69–75. Google Scholar | DOI

[7] [7] McDuff, D., The symplectic structure ofKàhlermanifolds of nonpositive curvature. J. Differential Geom. 28(1988), no. 3, 467–475. Google Scholar

[8] [8] McDuff, D. and Salamon, D., Introduction to symplectic topology. Second éd.,Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Google Scholar

[9] [9] Meigniez, G., Submersions, fibrations and bundles. Trans. Amer. Math. Soc. 354(2002), no. 9, 3771–3787. Google Scholar | DOI

[10] [10] Weinstein, A., Symplecticgroupoids and Poisson manifolds. Bull. Amer. Math.Soc. (N.S.) 16(1987), no. 1, 101–104. Google Scholar | DOI

Cité par Sources :