Proper groupoids and momentum maps : linearization, affinity, and convexity
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 39 (2006) no. 5, p. 841-869
@article{ASENS_2006_4_39_5_841_0,
     author = {Zung, Nguyen Tien},
     title = {Proper groupoids and momentum maps : linearization, affinity, and convexity},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Elsevier},
     volume = {Ser. 4, 39},
     number = {5},
     year = {2006},
     pages = {841-869},
     doi = {10.1016/j.ansens.2006.09.002},
     zbl = {1163.22001},
     zbl = {05137698},
     mrnumber = {2292634},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2006_4_39_5_841_0}
}
Zung, Nguyen Tien. Proper groupoids and momentum maps : linearization, affinity, and convexity. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 39 (2006) no. 5, pp. 841-869. doi : 10.1016/j.ansens.2006.09.002. http://www.numdam.org/item/ASENS_2006_4_39_5_841_0/

[1] Alekseev A., Malkin A., Meinrenken E., Lie group valued moment maps, J. Differential Geom. 48 (3) (1998) 445-495. | MR 1638045 | Zbl 0948.53045

[2] Bochner S., Compact groups of differentiable transformations, Ann. of Math. (2) 46 (1945) 372-381. | MR 13161 | Zbl 0063.00487

[3] Bursztyn H., Crainic M., Weinstein A., Zhu C., Integration of twisted Dirac brackets, Duke Math. J. 123 (3) (2004) 549-607. | MR 2068969 | Zbl 1067.58016

[4] Condevaux M., Dazord P., Molino P., Géométrie du moment, in: Séminaire Sud-Rhodanien I, Publications du Département de Mathématiques, Univ. Claude Bernard, Lyon I, 1988, pp. 131-160. | MR 1040871

[5] Coste A., Dazord P., Weinstein A., Groupoïdes symplectiques, in: 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, pp. 1-62. | Numdam | MR 996653 | Zbl 0668.58017

[6] Crainic M., Fernandes R.L., Integrability of Lie brackets, Ann. of Math. (2) 157 (2) (2003) 575-620. | MR 1973056 | Zbl 1037.22003

[7] Dufour J.-P., Zung N.T., Poisson Structures and Their Normal Forms, Progr. Math., vol. 242, Birkhäuser, Basel, 2005. | MR 2178041 | Zbl 1082.53078

[8] Duistermaat J.J., Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution, Trans. Am. Math. Soc. 275 (1983) 417-429. | MR 678361 | Zbl 0504.58020

[9] Flaschka H., Ratiu T., A convexity theorem for Poisson actions of compact Lie groups, Ann. Sci. École Norm. Sup. (4) 29 (6) (1996) 787-809. | Numdam | MR 1422991 | Zbl 0877.58025

[10] Giacobbe A., Convexity and multivalued Hamiltonians, Russ. Math. Surv. 55 (3) (2000) 578-580. | MR 1777360 | Zbl 1027.53097

[11] Ginzburg V.L., Weinstein A., Lie-Poisson structure on some Poisson Lie groups, J. Amer. Math. Soc. 5 (2) (1992) 445-453. | MR 1126117 | Zbl 0766.58018

[12] Grove K., Karcher H., Ruh E.A., Group actions and curvature, Invent. Math. 23 (1974) 31-48. | MR 385750 | Zbl 0271.53044

[13] Guillemin V., Sternberg S., Convexity properties of the moment mapping, Invent. Math. 67 (1982) 491-513. | MR 664117 | Zbl 0503.58017

[14] Haefliger A., Groupoïdes d'holonomie et classifiants, Astérisque 116 (1984) 70-97. | MR 755163 | Zbl 0562.57012

[15] Hilgert J., Neeb K.-H., Plank W., Symplectic convexity theorems and coadjoint orbits, Compos. Math. 94 (2) (1994) 129-180. | Numdam | MR 1302314 | Zbl 0819.22006

[16] Kirwan F., Convexity properties of the moment mapping. III, Invent. Math. 77 (3) (1984) 547-552. | MR 759257 | Zbl 0561.58016

[17] Knop F., Convexity of Hamiltonian manifolds, J. Lie Theory 12 (2) (2002) 571-582. | MR 1923787 | Zbl 1038.53080

[18] Koszul J.-L., Sur certains groupes de transformations de Lie, in: Géométrie différentielle. Colloques internationaux du CNRS, Strasbourg, 1953, CNRS, Paris, 1953, pp. 137-141. | MR 59919 | Zbl 0101.16201

[19] Lu J.-H., Momentum mappings and reduction of Poisson actions, in: 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 | Zbl 0735.58004

[20] Mikami K., Weinstein A., Moments and reduction for symplectic groupoids, Publ. Res. Inst. Math. Sci. 24 (1) (1988) 121-140. | MR 944869 | Zbl 0659.58016

[21] Mineur H., Sur les systèmes mécaniques dans lesquels figurent des paramètres fonctions du temps. Étude des systèmes admettant n intégrales premières uniformes en involution. Extension à ces systèmes des conditions de quantification de Bohr-Sommerfeld, Journal de l'École Polytechnique, Série III, 143ème année (1937) 173-191, 237-270. | Zbl 0017.04105

[22] Moerdijk I., Pronk D.A., Orbifolds, sheaves and groupoids, 12 (1) (1997) 3-21. | MR 1466622 | Zbl 0883.22005

[23] Moerdijk I., Mrcun J., Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003, ix+173 pp. | MR 2012261 | Zbl 1029.58012

[24] Molino P., Orbit-like foliations, in: Geometric Study of Foliations (Tokyo, 1993), World Scientific, River Edge, NJ, 1994, pp. 97-119. | MR 1363720

[25] Monnier Ph., Zung N.T., Levi decomposition for smooth Poisson structures, J. Differential Geom. 68 (2) (2004) 347-395. | MR 2144250 | Zbl 1085.53074

[26] Palais R.S., On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961) 295-323. | MR 126506 | Zbl 0103.01802

[27] Sjamaar R., Convexity properties of the moment mapping re-examined, Adv. Math. 138 (1) (1998) 46-91. | MR 1645052 | Zbl 0915.58036

[28] Weinstein A., Almost invariant submanifolds for compact group actions, J. Eur. Math. Soc. (JEMS) 2 (1) (2000) 53-86. | MR 1750452 | Zbl 0957.53021

[29] Weinstein A., Linearization problems for Lie algebroids and Lie groupoids, Lett. Math. Phys. 52 (1) (2000) 93-102. | MR 1800493 | Zbl 0961.22004

[30] Weinstein A., Poisson geometry of discrete series orbits, and momentum convexity for noncompact group actions, Lett. Math. Phys. 56 (1) (2001) 17-30. | MR 1848163 | Zbl 1010.53060

[31] Weinstein A., Linearization of regular proper groupoids, J. Inst. Math. Jussieu 1 (3) (2002) 493-511. | MR 1956059 | Zbl 1043.58009

[32] Xu P., Momentum maps and Morita equivalence, J. Diff. Geom. 67 (2) (2004) 289-333. | MR 2153080 | Zbl 1106.53057

[33] Zung N.T., Levi decomposition of analytic Poisson structures and Lie algebroids, Topology 42 (6) (2003) 1403-1420. | MR 1981362 | Zbl 1033.53069