Nous introduisons la classe modulaire d’une application de Poisson. Nous regardons quelques exemples et nous utilisons les classes modulaires des applications de Poisson pour étudier le comportement de la classe modulaire d’une variété de Poisson sous différents types de réduction. Nous discutons également leur version pour les groupoïdes symplectiques, qui prend ses valeurs dans la cohomologie du groupoïde.
We introduce the modular class of a Poisson map. We look at several examples and we use the modular classes of Poisson maps to study the behavior of the modular class of a Poisson manifold under different kinds of reduction. We also discuss their symplectic groupoid version, which lives in groupoid cohomology.
@article{AIF_2013__63_4_1285_0,
author = {Caseiro, Raquel and Fernandes, Rui Loja},
title = {The modular class of a Poisson map},
journal = {Annales de l'Institut Fourier},
volume = {63},
year = {2013},
pages = {1285-1329},
doi = {10.5802/aif.2804},
zbl = {06359590},
mrnumber = {3137356},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2013__63_4_1285_0}
}
Caseiro, Raquel; Fernandes, Rui Loja. The modular class of a Poisson map. Annales de l'Institut Fourier, Tome 63 (2013) pp. 1285-1329. doi : 10.5802/aif.2804. http://gdmltest.u-ga.fr/item/AIF_2013__63_4_1285_0/
[1] A brief introduction to Dirac manifolds (Preprint arXiv:1112.5037)
[2] Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model, Lett. Math. Phys., Tome 69 (2004), pp. 157-175 | Article | MR 2104442 | Zbl 1065.53063
[3] 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
[4] Integrability of Poisson brackets, J. Differential Geom., Tome 66 (2004) no. 1, pp. 71-137 http://projecteuclid.org/getRecord?id=euclid.jdg/1090415030 | MR 2128714 | Zbl 1066.53131
[5] Stability of symplectic leaves, Invent. Math., Tome 180 (2010) no. 3, pp. 481-533 | Article | MR 2609248 | Zbl 1197.53108
[6] Lectures on integrability of Lie brackets, Lectures on Poisson geometry, Geom. Topol. Publ., Coventry (Geom. Topol. Monogr.) Tome 17 (2011), pp. 1-107 | MR 2795150 | Zbl 1227.22005
[7] Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. (2), Tome 50 (1999) no. 200, pp. 417-436 | Article | MR 1726784 | Zbl 0968.58014
[8] Connections in Poisson geometry. I. Holonomy and invariants, J. Differential Geom., Tome 54 (2000) no. 2, pp. 303-365 http://projecteuclid.org/getRecord?id=euclid.jdg/1214341648 | MR 1818181 | Zbl 1036.53060
[9] Lie algebroids, holonomy and characteristic classes, Adv. Math., Tome 170 (2002) no. 1, pp. 119-179 | Article | MR 1929305 | Zbl 1007.22007
[10] The symplectization functor, XV International Workshop on Geometry and Physics, R. Soc. Mat. Esp., Madrid (Publ. R. Soc. Mat. Esp.) Tome 11 (2007), pp. 67-82 | MR 2504211 | Zbl 1229.53086
[11] Integrability of Poisson-Lie group actions, Lett. Math. Phys., Tome 90 (2009) no. 1-3, pp. 137-159 | Article | MR 2565037 | Zbl 1183.53075
[12] The momentum map in Poisson geometry, Amer. J. Math., Tome 131 (2009) no. 5, pp. 1261-1310 | Article | MR 2555841 | Zbl 1180.53083
[13] Equivariant Poisson cohomology and a spectral sequence associated with a moment map, Internat. J. Math., Tome 10 (1999) no. 8, pp. 977-1010 | Article | MR 1739368 | Zbl 1061.53059
[14] Holonomy on Poisson manifolds and the modular class, Israel J. Math., Tome 122 (2001), pp. 221-242 | Article | MR 1826501 | Zbl 0991.53055
[15] Poisson cohomology of Morita-equivalent Poisson manifolds, Internat. Math. Res. Notices (1992) no. 10, pp. 199-205 | Article | MR 1191570 | Zbl 0783.58026
[16] Homology and modular classes of Lie algebroids, Ann. Inst. Fourier (Grenoble), Tome 56 (2006) no. 1, pp. 69-83 http://aif.cedram.org/item?id=AIF_2006__56_1_69_0 | Article | Numdam | MR 2228680 | Zbl 1141.17018
[17] Modular classes of Lie algebroid morphisms, Transform. Groups, Tome 13 (2008) no. 3-4, pp. 727-755 | Article | MR 2452613 | Zbl 1167.53068
[18] Relative modular classes of Lie algebroids, C. R. Math. Acad. Sci. Paris, Tome 341 (2005) no. 8, pp. 509-514 | Article | MR 2180819 | Zbl 1080.22001
[19] Crochet de Schouten-Nijenhuis et cohomologie, Astérisque (1985) no. Numéro Hors Série, pp. 257-271 (The mathematical heritage of Élie Cartan (Lyon, 1984)) | MR 837203 | Zbl 0615.58029
[20] Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry, Tome 12 (1977) no. 2, pp. 253-300 | MR 501133 | Zbl 0405.53024
[21] Multiplicative and affine Poisson structures on Lie groups, ProQuest LLC, Ann Arbor, MI (1990) (Thesis (Ph.D.)–University of California, Berkeley) | MR 2685337
[22] Momentum mappings and reduction of Poisson actions, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), Springer, New York (Math. Sci. Res. Inst. Publ.) Tome 20 (1991), pp. 209-226 | Article | MR 1104930 | Zbl 0735.58004
[23] On the integrability of Lie subalgebroids, Adv. Math., Tome 204 (2006) no. 1, pp. 101-115 | Article | MR 2233128 | Zbl 1131.58015
[24] A classification of topologically stable Poisson structures on a compact oriented surface, J. Symplectic Geom., Tome 1 (2002) no. 3, pp. 523-542 http://projecteuclid.org/getRecord?id=euclid.jsg/1092403031 | Article | MR 1959058 | Zbl 1093.53087
[25] Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, Tome 40 (1988) no. 4, pp. 705-727 | Article | MR 959095 | Zbl 0642.58025
[26] The modular automorphism group of a Poisson manifold, J. Geom. Phys., Tome 23 (1997) no. 3-4, pp. 379-394 | Article | MR 1484598 | Zbl 0902.58013
[27] Dirac submanifolds and Poisson involutions, Ann. Sci. École Norm. Sup. (4), Tome 36 (2003) no. 3, pp. 403-430 | Article | Numdam | MR 1977824 | Zbl 1047.53052