AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Lectures on Poisson Geometry
About this Title
Marius Crainic, Utrecht University, Utrecht, The Netherlands, Rui Loja Fernandes, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL and Ioan Mărcuţ, Radboud University, Nijmegen, The Netherlands
Publication: Graduate Studies in Mathematics
Publication Year:
2021; Volume 217
ISBNs: 978-1-4704-6430-1 (print); 978-1-4704-6666-4 (online)
DOI: https://doi.org/10.1090/gsm/217
Table of Contents
Download chapters as PDF
Front/Back Matter
Basic concepts
- Poisson brackets
- Poisson bivectors
- Local structure of Poisson manifolds
- Notes and references for Part 1
Poisson geometry around leaves
- Symplectic leaves and the symplectic foliation
- Poisson transversals
- Symplectic realizations
- Dirac geometry
- Submanifolds in Poisson geometry
- Notes and references for Part 2
Global aspects
- Poisson cohomology
- Poisson homotopy
- Contravariant geometry and connections
- Notes and references for Part 3
Symplectic groupoids
- Complete symplectic realizations
- A crash course on Lie groupoids
- Symplectic groupoids
- Notes and references for Part 4
Appendices
- F. Alcalde-Cuesta and G. Hector, Intégration symplectique des variétés de Poisson régulières, Israel J. Math. 90 (1995), no. 1-3, 125–165 (French, with English summary). MR 1336320, DOI 10.1007/BF02783210
- F. Alcalde Cuesta and G. Hector, Feuilletages en surfaces, cycles évanouissants et variétés de Poisson, Monatsh. Math. 124 (1997), no. 3, 191–213 (French, with English summary). MR 1476362, DOI 10.1007/BF01298244
- Fernando Alcalde-Cuesta, Pierre Dazord, and Gilbert Hector, Sur l’intégration symplectique de la structure de Poisson singulière $\Lambda =(x^2+y^2)\partial /\partial x\wedge \partial /\partial y$ de $\textbf {R}^2$, Publ. Mat. 33 (1989), no. 3, 411–415 (French, with English summary). MR 1038479, DOI 10.5565/PUBLMAT_{3}3389_{0}2
- A. Alekseev and E. Meinrenken, On the Kashiwara-Vergne conjecture, Invent. Math. 164 (2006), no. 3, 615–634. MR 2221133, DOI 10.1007/s00222-005-0486-4
- A. Yu. Alekseev, On Poisson actions of compact Lie groups on symplectic manifolds, J. Differential Geom. 45 (1997), no. 2, 241–256. MR 1449971
- Anton Alekseev, Anton Malkin, and Eckhard Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998), no. 3, 445–495. MR 1638045
- Anton Alekseev and Eckhard Meinrenken, Linearization of Poisson Lie group structures, J. Symplectic Geom. 14 (2016), no. 1, 227–267. MR 3523256, DOI 10.4310/JSG.2016.v14.n1.a9
- Anton Alekseev and Charles Torossian, The Kashiwara-Vergne conjecture and Drinfeld’s associators, Ann. of Math. (2) 175 (2012), no. 2, 415–463. MR 2877064, DOI 10.4007/annals.2012.175.2.1
- Rui Almeida and Pierre Molino, Suites d’Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 1, 13–15 (French, with English summary). MR 778785
- Iakovos Androulidakis and Georges Skandalis, The holonomy groupoid of a singular foliation, J. Reine Angew. Math. 626 (2009), 1–37. MR 2492988, DOI 10.1515/CRELLE.2009.001
- Camilo Arias Abad and Marius Crainic, The Weil algebra and the Van Est isomorphism, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 3, 927–970 (English, with English and French summaries). MR 2918722, DOI 10.5802/aif.2633
- V. I. Arnol′d, Remarks on Poisson structures on a plane and on other powers of volume elements, Trudy Sem. Petrovsk. 12 (1987), 37–46, 242 (Russian, with English summary); English transl., J. Soviet Math. 47 (1989), no. 3, 2509–2516. MR 933050, DOI 10.1007/BF01102994
- Michèle Audin, Spinning tops, Cambridge Studies in Advanced Mathematics, vol. 51, Cambridge University Press, Cambridge, 1996. A course on integrable systems. MR 1409362
- Mélanie Bertelson, A $h$-principle for open relations invariant under foliated isotopies, J. Symplectic Geom. 1 (2002), no. 2, 369–425. MR 1959586
- K. H. Bhaskara and K. Viswanath, Poisson algebras and Poisson manifolds, Pitman Research Notes in Mathematics Series, vol. 174, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. MR 960879
- P. P. Boalch, Stokes matrices, Poisson Lie groups and Frobenius manifolds, Invent. Math. 146 (2001), no. 3, 479–506. MR 1869848, DOI 10.1007/s002220100170
- Lilian C. Brambila, Pedro Frejlich, and David Martínez Torres, Coregular submanifolds and poisson submersions, 2020, arXiv:2010.09058.
- Robert L. Bryant, Bochner-Kähler metrics, J. Amer. Math. Soc. 14 (2001), no. 3, 623–715. MR 1824987, DOI 10.1090/S0894-0347-01-00366-6
- Jean-Luc Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93–114. MR 950556
- Henrique Bursztyn, Semiclassical geometry of quantum line bundles and Morita equivalence of star products, Int. Math. Res. Not. 16 (2002), 821–846. MR 1891209, DOI 10.1155/S1073792802108014
- Henrique Bursztyn, A brief introduction to Dirac manifolds, Geometric and topological methods for quantum field theory, Cambridge Univ. Press, Cambridge, 2013, pp. 4–38. MR 3098084
- Henrique Bursztyn and Alejandro Cabrera, Multiplicative forms at the infinitesimal level, Math. Ann. 353 (2012), no. 3, 663–705. MR 2923945, DOI 10.1007/s00208-011-0697-5
- Henrique Bursztyn, Alejandro Cabrera, and Cristián Ortiz, Linear and multiplicative 2-forms, Lett. Math. Phys. 90 (2009), no. 1-3, 59–83. MR 2565034, DOI 10.1007/s11005-009-0349-9
- Henrique Bursztyn and Thiago Drummond, Lie theory of multiplicative tensors, Math. Ann. 375 (2019), no. 3-4, 1489–1554. MR 4023383, DOI 10.1007/s00208-019-01881-w
- Henrique Bursztyn, Hudson Lima, and Eckhard Meinrenken, Splitting theorems for Poisson and related structures, J. Reine Angew. Math. 754 (2019), 281–312. MR 4000576, DOI 10.1515/crelle-2017-0014
- Damien Calaque, Tony Pantev, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi, Shifted Poisson structures and deformation quantization, J. Topol. 10 (2017), no. 2, 483–584. MR 3653319, DOI 10.1112/topo.12012
- Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 1732868, DOI 10.1090/gsm/023
- Alberto Candel and Lawrence Conlon, Foliations. II, Graduate Studies in Mathematics, vol. 60, American Mathematical Society, Providence, RI, 2003. MR 1994394, DOI 10.1090/gsm/060
- Ana Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001. MR 1853077, DOI 10.1007/978-3-540-45330-7
- Ana Cannas da Silva and Alan Weinstein, Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, vol. 10, American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999. MR 1747916
- A. S. Cattaneo and M. Zambon, Coisotropic embeddings in Poisson manifolds, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3721–3746. MR 2491897, DOI 10.1090/S0002-9947-09-04667-4
- Alberto S. Cattaneo and Giovanni Felder, Poisson sigma models and symplectic groupoids, Quantization of singular symplectic quotients, Progr. Math., vol. 198, Birkhäuser, Basel, 2001, pp. 61–93. MR 1938552
- M. Condevaux, P. Dazord, and P. Molino, Géométrie du moment, Travaux du Séminaire Sud-Rhodanien de Géométrie, I, Publ. Dép. Math. Nouvelle Sér. B, vol. 88, Univ. Claude-Bernard, Lyon, 1988, pp. 131–160 (French). MR 1040871
- Jack F. Conn, Normal forms for analytic Poisson structures, Ann. of Math. (2) 119 (1984), no. 3, 577–601. MR 744864, DOI 10.2307/2007086
- Jack F. Conn, Normal forms for smooth Poisson structures, Ann. of Math. (2) 121 (1985), no. 3, 565–593. MR 794374, DOI 10.2307/1971210
- Ivan Contreras and Rui Loja Fernandes, Genus Integration, Abelianization, and Extended Monodromy, International Mathematics Research Notices (2019), rnz133.
- A. Coste, P. Dazord, and A. Weinstein, Groupoïdes symplectiques, Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A, vol. 87, Univ. Claude-Bernard, Lyon, 1987, pp. i–ii, 1–62 (French). MR 996653
- Ted Courant and Alan Weinstein, Beyond Poisson structures, Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986) Travaux en Cours, vol. 27, Hermann, Paris, 1988, pp. 39–49. MR 951168
- Theodore James Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631–661. MR 998124, DOI 10.1090/S0002-9947-1990-0998124-1
- Marius Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), no. 4, 681–721. MR 2016690, DOI 10.1007/s00014-001-0766-9
- Marius Crainic and Rui Loja Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620. MR 1973056, DOI 10.4007/annals.2003.157.575
- Marius Crainic and Rui Loja Fernandes, Integrability of Poisson brackets, J. Differential Geom. 66 (2004), no. 1, 71–137. MR 2128714
- Marius Crainic and Rui Loja Fernandes, Stability of symplectic leaves, Invent. Math. 180 (2010), no. 3, 481–533. MR 2609248, DOI 10.1007/s00222-010-0235-1
- Marius Crainic and Rui Loja Fernandes, A geometric approach to Conn’s linearization theorem, Ann. of Math. (2) 173 (2011), no. 2, 1121–1139. MR 2776372, DOI 10.4007/annals.2011.173.2.14
- Marius Crainic and Rui Loja Fernandes, Lectures on integrability of Lie brackets, Lectures on Poisson geometry, Geom. Topol. Monogr., vol. 17, Geom. Topol. Publ., Coventry, 2011, pp. 1–107. MR 2795150, DOI 10.2140/gt
- Marius Crainic, Rui Loja Fernandes, and David Martínez Torres, Poisson manifolds of compact types (PMCT 1), J. Reine Angew. Math. 756 (2019), 101–149. MR 4026450, DOI 10.1515/crelle-2017-0006
- Marius Crainic, Rui Loja Fernandes, and David Martínez Torres, Regular Poisson manifolds of compact types, Astérisque 413 (2019), viii + 154 (English, with English and French summaries). MR 4033498, DOI 10.24033/ast
- Marius Crainic and Ioan Mărcuţ, On the existence of symplectic realizations, J. Symplectic Geom. 9 (2011), no. 4, 435–444. MR 2900786
- Marius Crainic and Ioan Mărcuţ, A normal form theorem around symplectic leaves, J. Differential Geom. 92 (2012), no. 3, 417–461. MR 3005059
- I. Cruz and T. Fardilha, On sufficient and necessary conditions for linearity of the transverse Poisson structure, J. Geom. Phys. 60 (2010), no. 3, 543–551. MR 2600014, DOI 10.1016/j.geomphys.2009.12.001
- P. Dazord and G. Hector, Intégration symplectique des variétés de Poisson totalement asphériques, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 20, Springer, New York, 1991, pp. 37–72 (French, with English summary). MR 1104919, DOI 10.1007/978-1-4613-9719-9_{4}
- Pierre Dazord, Groupoïde d’holonomie et géométrie globale, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 1, 77–80 (French, with English and French summaries). MR 1435591, DOI 10.1016/S0764-4442(97)80107-3
- Pierre Dazord and Thomas Delzant, Le problème général des variables actions-angles, J. Differential Geom. 26 (1987), no. 2, 223–251 (French). MR 906389
- Matias del Hoyo and Daniel López Garcia, On Hausdorff integrations of Lie algebroids, Monatsh. Math. 194 (2021), no. 4, 811–833. MR 4228549, DOI 10.1007/s00605-021-01535-7
- Adrien Douady and Michel Lazard, Espaces fibrés en algèbres de Lie et en groupes, Invent. Math. 1 (1966), 133–151 (French). MR 197622, DOI 10.1007/BF01389725
- V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
- Pedro Duarte, Rui L. Fernandes, and Waldyr M. Oliva, Dynamics of the attractor in the Lotka-Volterra equations, J. Differential Equations 149 (1998), no. 1, 143–189. MR 1643678, DOI 10.1006/jdeq.1998.3443
- J.-P. Dufour, Linéarisation de certaines structures de Poisson, J. Differential Geom. 32 (1990), no. 2, 415–428 (French, with English summary). MR 1072912
- Jean-Paul Dufour and Nguyen Tien Zung, Poisson structures and their normal forms, Progress in Mathematics, vol. 242, Birkhäuser Verlag, Basel, 2005. MR 2178041, DOI 10.1007/3-7643-7335-0
- J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), no. 6, 687–706. MR 596430, DOI 10.1002/cpa.3160330602
- J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer-Verlag, Berlin, 2000. MR 1738431, DOI 10.1007/978-3-642-56936-4
- Charles Ehresmann, Introduction à la théorie des structures infinitésimales et des pseudogroupes de Lie, Colloque de topologie et géométrie différentielle, Strasbourg, 1952, no. 11, La Bibliothèque Nationale et Universitaire de Strasbourg, Strasbourg, 1953, pp. 16 (French). MR 0061454
- Sam Evens, Jiang-Hua Lu, and Alan Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 417–436. MR 1726784, DOI 10.1093/qjmath/50.200.417
- J. David Brown, Moody T. Chu, Donald C. Ellison, and Robert J. Plemmons (eds.), Proceedings of the Cornelius Lanczos International Centenary Conference, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. Held at North Carolina State University, Raleigh, North Carolina, December 12–17, 1993. MR 1298212
- Rui Loja Fernandes, Connections in Poisson geometry. I. Holonomy and invariants, J. Differential Geom. 54 (2000), no. 2, 303–365. MR 1818181
- Rui Loja Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), no. 1, 119–179. MR 1929305, DOI 10.1006/aima.2001.2070
- Rui Loja Fernandes and Pedro Frejlich, An $h$-principle for symplectic foliations, Int. Math. Res. Not. IMRN 7 (2012), 1505–1518. MR 2913182, DOI 10.1093/imrn/rnr077
- Rui Loja Fernandes and Pol Vanhaecke, Hyperelliptic Prym varieties and integrable systems, Comm. Math. Phys. 221 (2001), no. 1, 169–196. MR 1846906, DOI 10.1007/s002200100476
- Vladimir V. Fock and Alexander B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 6, 865–930 (English, with English and French summaries). MR 2567745, DOI 10.24033/asens.2112
- Philip Foth and Jiang-Hua Lu, Poisson structures on complex flag manifolds associated with real forms, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1705–1714. MR 2186993, DOI 10.1090/S0002-9947-05-03789-X
- Pedro Frejlich and Ioan Mărcuţ, The normal form theorem around Poisson transversals, Pacific J. Math. 287 (2017), no. 2, 371–391. MR 3632892, DOI 10.2140/pjm.2017.287.371
- Pedro Frejlich and Ioan Mărcuţ, On dual pairs in Dirac geometry, Math. Z. 289 (2018), no. 1-2, 171–200. MR 3803786, DOI 10.1007/s00209-017-1947-3
- Wee Liang Gan and Victor Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. 5 (2002), 243–255. MR 1876934, DOI 10.1155/S107379280210609X
- Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster algebras and Poisson geometry, Mathematical Surveys and Monographs, vol. 167, American Mathematical Society, Providence, RI, 2010. MR 2683456, DOI 10.1090/surv/167
- Viktor L. Ginzburg, Momentum mappings and Poisson cohomology, Internat. J. Math. 7 (1996), no. 3, 329–358. MR 1395934, DOI 10.1142/S0129167X96000207
- Viktor L. Ginzburg, Equivariant Poisson cohomology and a spectral sequence associated with a moment map, Internat. J. Math. 10 (1999), no. 8, 977–1010. MR 1739368, DOI 10.1142/S0129167X99000422
- Viktor L. Ginzburg, Grothendieck groups of Poisson vector bundles, J. Symplectic Geom. 1 (2001), no. 1, 121–169. MR 1959580
- Viktor L. Ginzburg and Alex Golubev, Holonomy on Poisson manifolds and the modular class, Israel J. Math. 122 (2001), 221–242. MR 1826501, DOI 10.1007/BF02809901
- Viktor L. Ginzburg and Jiang-Hua Lu, Poisson cohomology of Morita-equivalent Poisson manifolds, Internat. Math. Res. Notices 10 (1992), 199–205. MR 1191570, DOI 10.1155/S1073792892000229
- Viktor L. Ginzburg and Alan Weinstein, Lie-Poisson structure on some Poisson Lie groups, J. Amer. Math. Soc. 5 (1992), no. 2, 445–453. MR 1126117, DOI 10.1090/S0894-0347-1992-1126117-8
- Mark J. Gotay, On coisotropic imbeddings of presymplectic manifolds, Proc. Amer. Math. Soc. 84 (1982), no. 1, 111–114. MR 633290, DOI 10.1090/S0002-9939-1982-0633290-X
- Ryushi Goto, Rozansky-Witten invariants of log symplectic manifolds, Integrable systems, topology, and physics (Tokyo, 2000) Contemp. Math., vol. 309, Amer. Math. Soc., Providence, RI, 2002, pp. 69–84. MR 1953353, DOI 10.1090/conm/309/05342
- Marco Gualtieri, Generalized complex geometry, Ann. of Math. (2) 174 (2011), no. 1, 75–123. MR 2811595, DOI 10.4007/annals.2011.174.1.3
- Victor Guillemin, Eva Miranda, and Ana Rita Pires, Symplectic and Poisson geometry on $b$-manifolds, Adv. Math. 264 (2014), 864–896. MR 3250302, DOI 10.1016/j.aim.2014.07.032
- Victor W. Guillemin and Shlomo Sternberg, Remarks on a paper of Hermann, Trans. Amer. Math. Soc. 130 (1968), 110–116. MR 217226, DOI 10.1090/S0002-9947-1968-0217226-9
- Thomas Hawkins, Jacobi and the birth of Lie’s theory of groups, Arch. Hist. Exact Sci. 42 (1991), no. 3, 187–278. MR 1124967, DOI 10.1007/BF00375135
- Thomas Hawkins, Emergence of the theory of Lie groups, Sources and Studies in the History of Mathematics and Physical Sciences, Springer-Verlag, New York, 2000. An essay in the history of mathematics 1869–1926. MR 1771134, DOI 10.1007/978-1-4612-1202-7
- G. Hector, Une nouvelle obstruction à l’intégrabilité des variétés de Poisson régulières, Hokkaido Math. J. 21 (1992), no. 1, 159–185 (French, with English summary). MR 1153760, DOI 10.14492/hokmj/1381413261
- G. Hector, E. Macías, and M. Saralegi, Lemme de Moser feuilleté et classification des variétés de Poisson régulières, Publ. Mat. 33 (1989), no. 3, 423–430 (French, with English summary). MR 1038481, DOI 10.5565/PUBLMAT_{3}3389_{0}4
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original. MR 1834454, DOI 10.1090/gsm/034
- Josef Hofbauer and Karl Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998. MR 1635735, DOI 10.1017/CBO9781139173179
- Johannes Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57–113. MR 1058984, DOI 10.1515/crll.1990.408.57
- Vladimir Itskov, Mikhail Karasev, and Yurii Vorobjev, Infinitesimal Poisson cohomology, Coherent transform, quantization, and Poisson geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 187, Amer. Math. Soc., Providence, RI, 1998, pp. 327–360. MR 1728670, DOI 10.1090/trans2/187/03
- M. V. Karasëv, Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 3, 508–538, 638 (Russian). MR 854594
- M. V. Karasëv, Poisson algebras of symmetries and asymptotic behavior of spectral series, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 21–32, 96 (Russian). MR 831045
- M. V. Karasev and V. P. Maslov, Global asymptotic operators of a regular representation, Dokl. Akad. Nauk SSSR 257 (1981), no. 1, 33–37 (Russian). MR 608113
- A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001
- A. A. Kirillov, Local Lie algebras, Uspehi Mat. Nauk 31 (1976), no. 4(190), 57–76 (Russian). MR 0438390
- Maxim Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216. MR 2062626, DOI 10.1023/B:MATH.0000027508.00421.bf
- Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups. Part I, Mathematical Surveys and Monographs, vol. 56, American Mathematical Society, Providence, RI, 1998. MR 1614943, DOI 10.1090/surv/056
- Yvette Kosmann-Schwarzbach, La géométrie de Poisson, création du XXe siècle, Siméon-Denis Poisson, Hist. Math. Sci. Phys., Ed. Éc. Polytech., Palaiseau, 2013, pp. 129–172 (French, with French summary). MR 3202688
- Yvette Kosmann-Schwarzbach (ed.), Siméon-Denis Poisson, Histoire des Mathématiques et des Sciences Physiques. [History of Mathematics and Physics], Éditions de l’École Polytechnique, Palaiseau, 2013 (French). Les mathématiques au service de la science. [Mathematics in the service of science]. MR 3183750
- Bertram Kostant, Orbits, symplectic structures and representation theory, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) Nippon Hyoronsha, Tokyo, 1966, pp. p. 71. MR 0213476
- Jean-Louis Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque Numéro Hors Série (1985), 257–271 (French). The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837203
- Camille Laurent-Gengoux, Anne Pichereau, and Pol Vanhaecke, Poisson structures, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 347, Springer, Heidelberg, 2013. MR 2906391, DOI 10.1007/978-3-642-31090-4
- John M. Lee, Introduction to smooth manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. MR 2954043
- Songhao Li and Dylan Rupel, Symplectic groupoids for cluster manifolds, J. Geom. Phys. 154 (2020), 103688, 32. MR 4099478, DOI 10.1016/j.geomphys.2020.103688
- Paulette Libermann, Problèmes d’équivalence et géométrie symplectique, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982) Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 43–68 (French). MR 753129
- André Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), no. 2, 253–300 (French). MR 501133
- Jiang-Hua Lu, Momentum mappings and reduction of Poisson actions, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 20, Springer, New York, 1991, pp. 209–226. MR 1104930, DOI 10.1007/978-1-4613-9719-9_{1}5
- Jiang-Hua Lu and Alan Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501–526. MR 1037412
- Jiang-Hua Lu and Shizhuo Yu, Bott-Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces, Selecta Math. (N.S.) 26 (2020), no. 5, Paper No. 70, 61. MR 4160948, DOI 10.1007/s00029-020-00595-1
- K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, vol. 124, Cambridge University Press, Cambridge, 1987. MR 896907, DOI 10.1017/CBO9780511661839
- Kirill C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, vol. 213, Cambridge University Press, Cambridge, 2005. MR 2157566, DOI 10.1017/CBO9781107325883
- Kirill C. H. Mackenzie and Ping Xu, Integration of Lie bialgebroids, Topology 39 (2000), no. 3, 445–467. MR 1746902, DOI 10.1016/S0040-9383(98)00069-X
- Charles-Michel Marle, The inception of symplectic geometry: the works of Lagrange and Poisson during the years 1808–1810, Lett. Math. Phys. 90 (2009), no. 1-3, 3–21. MR 2565032, DOI 10.1007/s11005-009-0347-y
- David Martínez Torres, A note on the separability of canonical integrations of Lie algebroids, Math. Res. Lett. 17 (2010), no. 1, 69–75. MR 2592728, DOI 10.4310/MRL.2010.v17.n1.a6
- Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 3rd ed., Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2017. MR 3674984, DOI 10.1093/oso/9780198794899.001.0001
- Eckhard Meinrenken, Poisson geometry from a Dirac perspective, Lett. Math. Phys. 108 (2018), no. 3, 447–498. MR 3765970, DOI 10.1007/s11005-017-0977-4
- Kentaro Mikami and Alan Weinstein, Moments and reduction for symplectic groupoids, Publ. Res. Inst. Math. Sci. 24 (1988), no. 1, 121–140. MR 944869, DOI 10.2977/prims/1195175328
- Yoshihiko Mitsumatsu, Leafwise symplectic structures on Lawson’s foliation, J. Symplectic Geom. 16 (2018), no. 3, 817–838. MR 3882173, DOI 10.4310/JSG.2018.v16.n3.a6
- I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003. MR 2012261, DOI 10.1017/CBO9780511615450
- I. Moerdijk and J. Mrčun, On the universal enveloping algebra of a Lie algebroid, Proc. Amer. Math. Soc. 138 (2010), no. 9, 3135–3145. MR 2653938, DOI 10.1090/S0002-9939-10-10347-5
- Philippe Monnier, Poisson cohomology in dimension two, Israel J. Math. 129 (2002), 189–207. MR 1910942, DOI 10.1007/BF02773163
- Ioan Mărcuţ and Boris Osorno Torres, Deformations of log-symplectic structures, J. Lond. Math. Soc. (2) 90 (2014), no. 1, 197–212. MR 3245143, DOI 10.1112/jlms/jdu023
- Ryszard Nest and Boris Tsygan, Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems, Asian J. Math. 5 (2001), no. 4, 599–635. MR 1913813, DOI 10.4310/AJM.2001.v5.n4.a2
- Albert Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 390–397, 398–403. MR 0074879
- Yong-Geun Oh, Some remarks on the transverse Poisson structures of coadjoint orbits, Lett. Math. Phys. 12 (1986), no. 2, 87–91. MR 858269, DOI 10.1007/BF00416457
- Richard S. Palais, A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. 22 (1957), iii+123. MR 121424
- Tony Pantev, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi, Shifted symplectic structures, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 271–328. MR 3090262, DOI 10.1007/s10240-013-0054-1
- Manfred Plank, Hamiltonian structures for the $n$-dimensional Lotka-Volterra equations, J. Math. Phys. 36 (1995), no. 7, 3520–3534. MR 1339881, DOI 10.1063/1.530978
- Jean Pradines, Théorie de Lie pour les groupoïdes différentiables. Relations entre propriétés locales et globales, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A907–A910 (French). MR 214103
- Jean Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B 264 (1967), A245–A248 (French). MR 216409
- Jean Pradines, Troisième théorème de Lie les groupoïdes différentiables, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), A21–A23 (French). MR 231414
- J. A. Schouten, On the differential operators of first order in tensor calculus, Convegno Internazionale di Geometria Differenziale, Italia, 1953, Edizioni Cremonese, Roma, 1954, pp. 1–7. MR 0063750
- J.-M. Souriau, Quantification géométrique, Comm. Math. Phys. 1 (1966), 374–398 (French, with English summary). MR 207332
- Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532830
- Shlomo Sternberg, Lie algebras, University Press of Florida, 2009.
- Izu Vaisman, Remarks on the Lichnerowicz-Poisson cohomology, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 4, 951–963 (1991) (English, with French summary). MR 1096599
- Izu Vaisman, On the geometric quantization of Poisson manifolds, J. Math. Phys. 32 (1991), no. 12, 3339–3345. MR 1137387, DOI 10.1063/1.529446
- Izu Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Mathematics, vol. 118, Birkhäuser Verlag, Basel, 1994. MR 1269545, DOI 10.1007/978-3-0348-8495-2
- W. T. van Est, Une démonstration de É. Cartan du troisième théorème de Lie, Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986) Travaux en Cours, vol. 27, Hermann, Paris, 1988, pp. 83–96 (French). MR 951172
- Yu. M. Vorob′ev and M. V. Karasëv, Poisson manifolds and the Schouten bracket, Funktsional. Anal. i Prilozhen. 22 (1988), no. 1, 1–11, 96 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 1, 1–9. MR 936694, DOI 10.1007/BF01077717
- Yu. M. Vorob′ev and M. V. Karasëv, Deformation and cohomologies of Poisson brackets [in Topological and geometric methods of analysis (Russian), 75–89, Voronezh. Gos. Univ., Voronezh, 1989; MR1047671 (91g:58098)], Global analysis—studies and applications, IV, Lecture Notes in Math., vol. 1453, Springer, Berlin, 1990, pp. 271–289. MR 1096519, DOI 10.1007/BFb0085961
- Pavol Ševera, Some title containing the words “homotopy” and “symplectic”, e.g. this one, Travaux mathématiques. Fasc. XVI, Trav. Math., vol. 16, Univ. Luxemb., Luxembourg, 2005, pp. 121–137. MR 2223155
- Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman & Co., Glenview, Ill.-London, 1971. MR 0295244
- Alan Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), no. 3, 523–557. MR 723816
- Alan Weinstein, Errata and addenda: “The local structure of Poisson manifolds” [J. Differential Geom. 18 (1983), no. 3, 523–557; MR0723816 (86i:58059)], J. Differential Geom. 22 (1985), no. 2, 255. MR 834280
- Alan Weinstein, Poisson structures and Lie algebras, Astérisque Numéro Hors Série (1985), 421–434. The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837210
- Alan Weinstein, Poisson geometry of the principal series and nonlinearizable structures, J. Differential Geom. 25 (1987), no. 1, 55–73. MR 873455
- Alan Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 1, 101–104. MR 866024, DOI 10.1090/S0273-0979-1987-15473-5
- Alan Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), no. 4, 705–727. MR 959095, DOI 10.2969/jmsj/04040705
- Alan Weinstein, Blowing up realizations of Heisenberg-Poisson manifolds, Bull. Sci. Math. 113 (1989), no. 4, 381–406 (English, with French summary). MR 1029617
- Alan Weinstein, Noncommutative geometry and geometric quantization, Symplectic geometry and mathematical physics (Aix-en-Provence, 1990) Progr. Math., vol. 99, Birkhäuser Boston, Boston, MA, 1991, pp. 446–461. MR 1156554
- Alan Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), no. 3-4, 379–394. MR 1484598, DOI 10.1016/S0393-0440(97)80011-3
- Alan Weinstein, Poisson geometry, Differential Geom. Appl. 9 (1998), no. 1-2, 213–238. Symplectic geometry. MR 1636305, DOI 10.1016/S0926-2245(98)00022-9
- Alan Weinstein and Ping Xu, Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), 159–189. MR 1103911
- Ping Xu, Morita equivalence of Poisson manifolds, Comm. Math. Phys. 142 (1991), no. 3, 493–509. MR 1138048
- Ping Xu, Morita equivalence and symplectic realizations of Poisson manifolds, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 3, 307–333. MR 1169134
- Ping Xu, Poisson cohomology of regular Poisson manifolds, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 967–988 (English, with English and French summaries). MR 1196101
- Ping Xu, Dirac submanifolds and Poisson involutions, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 3, 403–430 (English, with English and French summaries). MR 1977824, DOI 10.1016/S0012-9593(03)00013-2