The modular class of a Poisson map

Raquel Caseiro[1]; Rui Loja Fernandes[2]

  • [1] Universidade de Coimbra CMUC, Department of Mathematics 3001-454 Coimbra, (Portugal)
  • [2] University of Illinois at Urbana-Champaign Department of Mathematics 1409 W. Green Street Urbana, IL 61801 (USA)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 4, page 1285-1329
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Caseiro, Raquel, and Fernandes, Rui Loja. "The modular class of a Poisson map." Annales de l’institut Fourier 63.4 (2013): 1285-1329. <http://eudml.org/doc/275504>.

@article{Caseiro2013,
abstract = {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.},
affiliation = {Universidade de Coimbra CMUC, Department of Mathematics 3001-454 Coimbra, (Portugal); University of Illinois at Urbana-Champaign Department of Mathematics 1409 W. Green Street Urbana, IL 61801 (USA)},
author = {Caseiro, Raquel, Fernandes, Rui Loja},
journal = {Annales de l’institut Fourier},
keywords = {Poisson manifold; Poisson map; modular class},
language = {eng},
number = {4},
pages = {1285-1329},
publisher = {Association des Annales de l’institut Fourier},
title = {The modular class of a Poisson map},
url = {http://eudml.org/doc/275504},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Caseiro, Raquel
AU - Fernandes, Rui Loja
TI - The modular class of a Poisson map
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 4
SP - 1285
EP - 1329
AB - 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.
LA - eng
KW - Poisson manifold; Poisson map; modular class
UR - http://eudml.org/doc/275504
ER -

References

top
  1. H. Bursztyn, A brief introduction to Dirac manifolds Zbl1293.53001
  2. Alberto S. Cattaneo, Giovanni Felder, Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model, Lett. Math. Phys. 69 (2004), 157-175 Zbl1065.53063MR2104442
  3. Marius Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), 681-721 Zbl1041.58007MR2016690
  4. Marius Crainic, Rui Loja Fernandes, Integrability of Poisson brackets, J. Differential Geom. 66 (2004), 71-137 Zbl1066.53131MR2128714
  5. Marius Crainic, Rui Loja Fernandes, Stability of symplectic leaves, Invent. Math. 180 (2010), 481-533 Zbl1197.53108MR2609248
  6. Marius Crainic, Rui Loja Fernandes, Lectures on integrability of Lie brackets, Lectures on Poisson geometry 17 (2011), 1-107, Geom. Topol. Publ., Coventry Zbl1227.22005MR2795150
  7. Sam Evens, Jiang-Hua Lu, Alan Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. (2) 50 (1999), 417-436 Zbl0968.58014MR1726784
  8. Rui Loja Fernandes, Connections in Poisson geometry. I. Holonomy and invariants, J. Differential Geom. 54 (2000), 303-365 Zbl1036.53060MR1818181
  9. Rui Loja Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), 119-179 Zbl1007.22007MR1929305
  10. Rui Loja Fernandes, The symplectization functor, XV International Workshop on Geometry and Physics 11 (2007), 67-82, R. Soc. Mat. Esp., Madrid Zbl1229.53086MR2504211
  11. Rui Loja Fernandes, David Iglesias Ponte, Integrability of Poisson-Lie group actions, Lett. Math. Phys. 90 (2009), 137-159 Zbl1183.53075MR2565037
  12. Rui Loja Fernandes, Juan-Pablo Ortega, Tudor S. Ratiu, The momentum map in Poisson geometry, Amer. J. Math. 131 (2009), 1261-1310 Zbl1180.53083MR2555841
  13. Viktor L. Ginzburg, Equivariant Poisson cohomology and a spectral sequence associated with a moment map, Internat. J. Math. 10 (1999), 977-1010 Zbl1061.53059MR1739368
  14. Viktor L. Ginzburg, Alex Golubev, Holonomy on Poisson manifolds and the modular class, Israel J. Math. 122 (2001), 221-242 Zbl0991.53055MR1826501
  15. Viktor L. Ginzburg, Jiang-Hua Lu, Poisson cohomology of Morita-equivalent Poisson manifolds, Internat. Math. Res. Notices (1992), 199-205 Zbl0783.58026MR1191570
  16. Janusz Grabowski, Giuseppe Marmo, Peter W. Michor, Homology and modular classes of Lie algebroids, Ann. Inst. Fourier (Grenoble) 56 (2006), 69-83 Zbl1141.17018MR2228680
  17. Yvette Kosmann-Schwarzbach, C. Laurent-Gengoux, Alan Weinstein, Modular classes of Lie algebroid morphisms, Transform. Groups 13 (2008), 727-755 Zbl1167.53068MR2452613
  18. Yvette Kosmann-Schwarzbach, Alan Weinstein, Relative modular classes of Lie algebroids, C. R. Math. Acad. Sci. Paris 341 (2005), 509-514 Zbl1080.22001MR2180819
  19. Jean-Louis Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque (1985), 257-271 Zbl0615.58029MR837203
  20. André Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), 253-300 Zbl0405.53024MR501133
  21. Jiang-Hua Lu, Multiplicative and affine Poisson structures on Lie groups, (1990), ProQuest LLC, Ann Arbor, MI MR2685337
  22. Jiang-Hua Lu, Momentum mappings and reduction of Poisson actions, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989) 20 (1991), 209-226, Springer, New York Zbl0735.58004MR1104930
  23. I. Moerdijk, J. Mrčun, On the integrability of Lie subalgebroids, Adv. Math. 204 (2006), 101-115 Zbl1131.58015MR2233128
  24. Olga Radko, A classification of topologically stable Poisson structures on a compact oriented surface, J. Symplectic Geom. 1 (2002), 523-542 Zbl1093.53087MR1959058
  25. Alan Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), 705-727 Zbl0642.58025MR959095
  26. Alan Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379-394 Zbl0902.58013MR1484598
  27. Ping Xu, Dirac submanifolds and Poisson involutions, Ann. Sci. École Norm. Sup. (4) 36 (2003), 403-430 Zbl1047.53052MR1977824

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.