# Poisson Book (2005)

Jean-Paul Dufour and Nguyen Tien Zung, Poisson structures and their normal forms, Progress in Mathematics, Vol. 242, Birkhäuser Verlag, Basel, 2005. xvi+321 pp.

The book has  been reviewed by Jan Sanders (University of Amsterdam) for LMS. His review can be found here:Bull. London Math. Soc.,  40 (2008), 1094-1095.

Jan Sanders says that the book is well-written, and that he recommends it to anyone “with a fair interest in the interaction between formal algebraic methods, analysis and physical applications”. I guess that includes almost every mathematician and physicist, so you may as well order the book if you don’t have it yet

The first chapters of this book  may serve as an introduction to Poisson geometry for interested people. They actually grew out from a graduate course that Jean-Paul and I taught in Montpellier around 2000.

The other chapters are about the normal forms of Poisson structures and related objects, including vector fields, singular foliations, Lie groupoids and Lie algebroids.

Some results in the book are original had have not been previously published elsewhere. These include generalized Kupka phenomena (in the chapter on Nambu structures and singular foliations – the ideas there area mainly due to Jean-Paul), and linearization problem for Poisson structures with a con-compact simisimple linear part (joint work with Philippe Monnier).

There are some small errors in a paragraph about quadratization of Poisson structures, pointed out to us by Laurent Stolovitch. If anyone is working on it, please consult with Laurent or Philipp Lohrmann who imporoved some of our quadratization results.

After the publication of the book, there have been new interesting results about normal forms of Poisson structures, including: Crainic, Fernandes, Martinez (geometric approach to linearization of Poisson structures and groupoids, Poisson structures of compact type), Marcut, Vorobiev (normal forms in a neighborhood of a symplectic leaf), Stolovitch and Lohrmann (analytic and Gevrey-class normal forms), Monnier, Miranda and myself (rigidity of Hamiltonian actions), and so on. Comments welcome, especially about new developments.

I’m glad to see that this book could serve many people as an introduction to Poisson structures and also to Lie algebroids and groupoids.

Cited in:

1. # M Crainic, On the linearization theorem for proper Lie groupoids, Arxiv preprint arXiv:1103.5245, 2011
2. # Pantelis A. Damianou, Fani Petalidou, Poisson brackets with prescribed Casimirs, arXiv:1103.0849, 2011.
3. # Dennise García-Beltrán, José A. Vallejo, Yurii Vorobjev, On Lie algebroids and Poisson algebras, arXiv:1106.1512 (2011).
4. # AF McMillan, On Embedding Singular Poisson Spaces, Arxiv preprint arXiv:1108.2207, 2011.
5. # M.J. Pflaum, H. Posthuma, X. Tang, Geometry of orbit spaces of proper Lie groupoids, arXiv:1101.0180 (last version: September 2011)
6. # M Crainic, On the existence of symplectic realizations, preprint arXiv:1009.2085, 2010.
7. # Amine Bahayou, Mohamed Boucetta, Multiplicative deformations of spectrale triples associated to left invariant metrics on Lie groups, preprint arXiv:0906.2887, 2009.
8. # H. Bursztyn, V. Dolgushev, S. Waldmann, Morita equivalence and characteristic classes of star products, arXiv:0909.4259 (2009)
9. M Boucetta, A class of Poisson structures compatible with the canonical Poisson structure on the cotangent bundle, Comptes Rendus Mathematique, Volume 349, Issues 5-6, March 2011, Pages 331-335.
10. Mohamed Boucetta, Alberto Medina, Solutions of the Yang–Baxter equations on quadratic Lie groups: The case of oscillator groups, Journal of Geometry and Physics, Volume 61, Issue 12, December 2011, Pages 2309-2320
11. O Brahic, C Zhu, Lie algebroid fibrations, Advances in Mathematics, Volume 226, Issue 4, 1 March 2011, Pages 3105-3135
12. Arlo Caine, Toric Poisson structures, Moscow Mathematical Journal, Vol .11 (2011), No. 2, 205-229.
13. ! ЮМ Воробьев, Полулокальные нормальные формы пуассоновых структур и гамильтонизация динамических систем, Doktor Nauk Dissertation, Moscow 2011.
14. G Trentinaglia, Some remarks on the global structure of proper Lie groupoids in low codimensions, Topology and its Applications, 2011.
15. JP Dufour, Decomposability of a Poisson tensor could be a stable phenomenon, Afrcian Diaspora Journal of Mathematics 9 (2010), No. 2, 47-81.
16. Viktoria Heu, Universal isomonodromic deformations of meromorphic rank 2 connections on curves, Annales de l’institut Fourier, 60 no. 2 (2010), p. 515-549.
17. P Lohrmann, Normalisation holomorphe de structures de Poisson, Ergodic Theory and Dynamical Systems, 2010.
18. F. Petalidou, On twisted contact groupoids and on integration of twisted Jacobi manifolds, Bulletin des Sciences Math., 2010.
19. Bahayou, Amine ; Boucetta, Mohamed . Metacurvature of Riemannian Poisson-Lie groups. J. Lie Theory 19 (2009), no. 3, 439–462.
20. Lohrmann, Philipp . Classification analytique de structures de Poisson. Bull. Soc. Math. France 137 (2009), no. 3, 321–386.
21. Crasmareanu, Mircea . Last multipliers for multivectors with applications to Poisson geometry. Taiwanese J. Math. 13 (2009), no. 5, 1623–1636.
22. Crasmareanu, Mircea ; Hreţcanu, Cristina-Elena . Last multipliers on Lie algebroids. Proc. Indian Acad. Sci. Math. Sci. 119 (2009), no. 3, 287–296.
23. Bahayou, Amine ; Boucetta, Mohamed . Multiplicative noncommutative deformations of left invariant Riemannian metrics on Heisenberg groups. C. R. Math. Acad. Sci. Paris 347 (2009), no. 13-14, 791–796.
24. Pelap, Serge Roméo Tagne . Poisson (co)homology of polynomial Poisson algebras in dimension four: Sklyanin’s case. J. Algebra 322 (2009), no. 4, 1151–1169.
25. Goto, Ryushi . Poisson structures and generalized Kähler submanifolds. J. Math. Soc. Japan 61 (2009), no. 1, 107–132.
26. L. Stolovitch, “Rigidity of Poisson Structures”, Особенности и приложения, Сборник статей, Тр. МИАН, 267, МАИК, М., 2009, 266–279. (Proceedings of the Steklov Institute of Mathematics).
27. Tagne Pelap, Serge Roméo . On the Hochschild homology of elliptic Sklyanin algebras. Lett. Math. Phys. 87 (2009), no. 3, 267–281.
28. Tudoran, Răzvan M. ; Tudoran, Ramona A. On a large class of three-dimensional Hamiltonian systems. J. Math. Phys. 50 (2009), no. 1, 012703, 9 pp.
29. Lawton, Sean . Poisson geometry of ${\rm SL}(3,\Bbb C)$-character varieties relative to a surface with boundary. Trans. Amer. Math. Soc. 361 (2009), no. 5, 2397–2429.
30. M. Bordemann, A. Makhlouf, Formality and deformation of universal enveloping algebras, International Journal of Theoretical Physics, 47 (2008), 311–332.
31. P. Cartier, Groupoides de Lie et leurs algébroides, Séminaire Bourbaki (2008)
32. JP Dufour, Examples of higher order stable singularities of Poisson structures, Contemporary Math. 450 (2008), 103–111.
33. Kosmann-Schwarzbach, Y. ; Laurent-Gengoux, C. ; Weinstein, A. Modular classes of Lie algebroid morphisms. Transform. Groups 13 (2008), no. 3-4, 727–755.
34. Lohrmann, Philipp . Sectorial normalization of Poisson structures. C. R. Math. Acad. Sci. Paris 346 (2008), no. 15-16, 829–832.
35. Damianou, Pantelis A. ; Fernandes, Rui Loja . Integrable hierarchies and the modular class. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 1, 107–137.
36. Lin, Qian ; Liu, Zhangju ; Sheng, Yunhe . Quadratic deformations of Lie-Poisson structures. Lett. Math. Phys. 83 (2008), no. 3, 217–229.
37. Fassò, Francesco ; Sansonetto, Nicola . Integrable almost-symplectic Hamiltonian systems. J. Math. Phys. 48 (2007), no. 9, 092902, 13 pp.
38. Ctirad Klimcik, $q \to \infty$ limit of the quasitriangular WZW model, J. Nonlinear Math. Phys. 14 (2007), No. 4, 494–526.
39. * M. Henkel, J. Unterberger, Supersymmetric extension of Schrodinger-invariance, Nuclear Physics B, vol. 746 (2006), no. 3, 155–201.
40. Dufour, Jean-Paul ; Wade, Aïssa . Stability of higher order singular points of Poisson manifolds and Lie algebroids. Ann. Inst. Fourier (Grenoble) 56 (2006), no. 3, 545–559.
41. Pichereau, Anne . Poisson (co)homology and isolated singularities. J. Algebra 299 (2006), no. 2, 747–777.
42. RL Fernandes, Ph Monnier, Linearization of Poisson beackets, Lett. Math. Phys. 69 (2004), 89–114.
43. ^ NT Zung, Entropy of geometric structures, Bulletin Brazilian Math Soc (2011).
44. ^ Zung, Nguyen Tien . Proper groupoids and momentum maps: linearization, affinity, and convexity. Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 5, 841–869.
45. ^ Miranda, Eva ; Zung, Nguyen Tien . A note on equivariant normal forms of Poisson structures. Math. Res. Lett. 13 (2006), no. 5-6, 1001–1012.
46. ^ Monnier, Philippe ; Zung, Nguyen Tien . Normal forms of vector fields on Poisson manifolds. Ann. Math. Blaise Pascal 13 (2006), no. 2, 349–380.