Integrability of Jacobi and Poisson structures
Annales de l'Institut Fourier, Volume 57 (2007) no. 4, p. 1181-1216
We discuss the integrability of Jacobi manifolds by contact groupoids, and then look at what the Jacobi point of view brings new into Poisson geometry. In particular, using contact groupoids, we prove a Kostant-type theorem on the prequantization of symplectic groupoids, which answers a question posed by Weinstein and Xu. The methods used are those of Crainic-Fernandes on A-paths and monodromy group(oid)s of algebroids. In particular, most of the results we obtain are valid also in the non-integrable case.
Nous discutons l’intégrabilité des variétés de Jacobi par des groupoïdes de contact. Nous considérons ensuite ce que le point de vue des structures de Jacobi apporte à la géométrie de Poisson. En particulier, en utilisant les groupoïdes de contacts, nous prouvons un théorème à la Kostant sur la préquantization des groupoïdes symplectiques. Ce théorème répond à une question posée par Weinstein et Xu. Nous utilisons les méthodes de Crainic-Fernandes sur les A-paths et les group(oïd)es de monodromie d’algebroïdes. En particulier, la plupart des résultats que nous obtenons sont valides dans le cas non-intégrable.
DOI : https://doi.org/10.5802/aif.2291
Classification:  53D17
Keywords: Jacobi structure, Poisson geometry, prequantization, contact groupoids, integration
@article{AIF_2007__57_4_1181_0,
     author = {Crainic, Marius and Zhu, Chenchang},
     title = {Integrability of Jacobi and Poisson structures},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {4},
     year = {2007},
     pages = {1181-1216},
     doi = {10.5802/aif.2291},
     zbl = {1146.53055},
     mrnumber = {2339329},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2007__57_4_1181_0}
}
Crainic, Marius; Zhu, Chenchang. Integrability of Jacobi and Poisson structures. Annales de l'Institut Fourier, Volume 57 (2007) no. 4, pp. 1181-1216. doi : 10.5802/aif.2291. https://aif.centre-mersenne.org/item/AIF_2007__57_4_1181_0/

[1] Blair, David E. Riemannian geometry of contact and symplectic manifolds, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 203 (2002) | MR 1874240 | Zbl 1011.53001

[2] Bursztyn, Henrique; Crainic, Marius; Weinstein, Alan; Zhu, Chenchang Integration of twisted Dirac brackets, Duke Math. J., Tome 123 (2004) no. 3, pp. 549-607 | Article | MR 2068969 | Zbl 1067.58016

[3] Cannas Da Silva, Ana; Weinstein, Alan Geometric models for noncommutative algebras, American Mathematical Society, Providence, RI, Berkeley Mathematics Lecture Notes, Tome 10 (1999) | MR 1747916 | Zbl 01515267

[4] Cattaneo, Alberto S.; Felder, Giovanni Poisson sigma models and symplectic groupoids, Quantization of singular symplectic quotients, Birkhäuser, Basel (Progr. Math.) Tome 198 (2001), pp. 61-93 | MR 1938552 | Zbl 1038.53074

[5] Crainic, Marius Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv., Tome 78 (2003) no. 4, pp. 681-721 | Article | MR 2016690 | Zbl 1041.58007

[6] Crainic, Marius; Fernandes, Rui Loja Integrability of Lie brackets, Ann. of Math. (2), Tome 157 (2003) no. 2, pp. 575-620 | Article | MR 1973056 | Zbl 1037.22003

[7] Crainic, Marius; Fernandes, Rui Loja Integrability of Poisson brackets, J. Differential Geom., Tome 66 (2004) no. 1, pp. 71-137 | MR 2128714 | Zbl 1066.53131

[8] Dazord, Pierre Intégration d’algèbres de Lie locales et groupoïdes de contact, C. R. Acad. Sci. Paris Sér. I Math., Tome 320 (1995) no. 8, pp. 959-964 | Zbl 0843.57036

[9] Dazord, Pierre Sur l’intégration des algèbres de Lie locales et la préquantification, Bull. Sci. Math., Tome 121 (1997) no. 6, pp. 423-462 | Zbl 1053.53523

[10] Dazord, Pierre; Lichnerowicz, André; Marle, Charles-Michel Structure locale des variétés de Jacobi, J. Math. Pures Appl. (9), Tome 70 (1991) no. 1, pp. 101-152 | MR 1091922 | Zbl 0659.53033

[11] Guedira, Fouzia; Lichnerowicz, André Géométrie des algèbres de Lie locales de Kirillov, J. Math. Pures Appl. (9), Tome 63 (1984) no. 4, pp. 407-484 | MR 789560 | Zbl 0562.53029

[12] Higgins, Philip J.; Mackenzie, Kirill Algebraic constructions in the category of Lie algebroids, J. Algebra, Tome 129 (1990) no. 1, pp. 194-230 | Article | MR 1037400 | Zbl 0696.22007

[13] Iglesias-Ponte, David; Marrero, Juan C. Jacobi groupoids and generalized Lie bialgebroids, J. Geom. Phys., Tome 48 (2003) no. 2-3, pp. 385-425 | Article | MR 2007602 | Zbl 1037.17023

[14] Kerbrat, Yvan; Souici-Benhammadi, Zoubida Variétés de Jacobi et groupoïdes de contact, C. R. Acad. Sci. Paris Sér. I Math., Tome 317 (1993) no. 1, pp. 81-86 | MR 1228970 | Zbl 0804.58015

[15] Kirillov, A. A. Local Lie algebras, Uspehi Mat. Nauk, Tome 31 (1976) no. 4(190), pp. 57-76 | MR 438390 | Zbl 0352.58014

[16] Lichnerowicz, André Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl. (9), Tome 57 (1978) no. 4, pp. 453-488 | MR 524629 | Zbl 0407.53025

[17] Moerdijk, Ieke; Mrčun, Janez On integrability of infinitesimal actions, Amer. J. Math., Tome 124 (2002) no. 3, pp. 567-593 | Article | MR 1902889 | Zbl 1013.58010

[18] Palais, Richard S. A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. No., Tome 22 (1957), pp. iii+123 | MR 121424 | Zbl 0178.26502

[19] Weinstein, Alan Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.), Tome 16 (1987) no. 1, pp. 101-104 | Article | MR 866024 | Zbl 0618.58020

[20] Weinstein, Alan; Xu, Ping Extensions of symplectic groupoids and quantization, J. Reine Angew. Math., Tome 417 (1991), pp. 159-189 | MR 1103911 | Zbl 0722.58021

[21] Zambon, Marco; Zhu, Chenchang Contact reduction and groupoid actions, Trans. Amer. Math. Soc., Tome 358 (2006) no. 3, p. 1365-1401 (electronic) | Article | MR 2187657 | Zbl 1092.53056