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Outline

Poisson Geometry and Applications

2000, Oberwolfach Reports

https://doi.org/10.4171/OWR/2007/23

Abstract

This workshop concentrated on some of the major areas of Poisson geometry and its applications.

References (107)

  1. C. Arias Abad, M. Crainic, Representations up to homotopy for Lie algebroids, in progress.
  2. K. Behrend, On the de Rham cohomology of differential and algebraic stacks, Advances in Mathematics 198 (2005), no 2, 583-622.
  3. R.Bott, On the Chern-Weil homomorphism and continuous cohomology of Lie groups, Ad- vances in Mathematics 11, 289-303 (1973).
  4. R.Bott, L.Tu Equivariant characteristic classes in the Cartan model, arXiv:math/0102001. preprint.
  5. E. Getzler, The equivariant chern character for noncompact groups, Advances in Mathe- matics 109, 88-117 (1994).
  6. V. Guillemin, S. Sternberg, Supersymmetry and equivariant de Rham theory, Springer- Verlag (1999).
  7. J. Kalkman, A BRST model for equivariant cohomology applied to symplectic geometry, PhD Thesis, Utrecht, 1993.
  8. I. Moerdijk, Orbifolds as groupoids: an introduction, Orbifolds in mathematics and physics(Madison, WI,2001),205-222, Contemp. Math, 310, Amer. Math Soc, Providence, RI,2002.
  9. D. Quillen, Superconnections and the Chern character, Topology 24 (1985), no.1, 89-95. References
  10. H. Bursztyn, G. R. Cavalcanti, and M. Gualtieri. Reduction of Courant algebroids and generalized complex structures, Adv. in Math. 211 (2007), 726-765.
  11. H. Bursztyn, G. R. Cavalcanti, and M. Gualtieri. Generalized Kähler and hiper-Kähler quotients, arXiv:math/0702104v1 [math.DG].
  12. References
  13. A. S. Cattaneo, Deformation quantization and reduction, math/0701378. Double of a cluster variety Vladimir Fock (joint work with Aleksandr Goncharov) References
  14. V.V. Fock, A.B. Goncharov Cluster X-varieties, amalgamation and Poisson-Lie groups, n Algebraic Geometry Theory and Number Theory, Birkhäuser, Progress in math. Vol. 253, 2006, arXiv:math.RT/0508408
  15. V.V. Fock, A.B. Goncharov The quantum dilogarithm and unitary representations of the cluster mapping class groups, arXiv:math/0702397
  16. M. Bordemann, M. Brischle, C. Emmrich, S. Waldmann, Phase Space Reduction for Star Products : An Explicit Construction for CP n , Lett. Math. Phys. 36 (1996), 357-371.
  17. P. Bieliavsky, M. Cahen, S. Gutt, J. Rawnsley, L. Schwachhöfer, Symplectic connections, preprint SG/0511194, International Journal of Geometric Methods in Modern Physics, Vol. 3, No. 3 (2006) 375-420
  18. Cahen M., Gutt S., Schwachhöfer L.: Construction of Ricci-type connections by reduction and induction, in The breadth of symplectic and Poisson Geometry, Marsden, J.E. and Ratiu, T.S. (eds), Progress in Math 232, Birkhauser, 2004, 41-57.
  19. Cahen M. and Schachhöfer L., Special symplectic connections, preprint DG0402221. References
  20. V. Fock, A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1-211.
  21. S. Fomin, A. Zelevinsky, Cluster algebras. I. Foundations J. Amer. Math. Soc. 15 no. 2 (2002), 497-529 (electronic).
  22. A. Henriques, J. Kamnitzer, Crystals and cobounday categories, Duke Math. J. 132 no. 2 (2006), 191-216.
  23. R. C. Penner, The decorated Teichmüller space of punctured surfaces Comm. Math. Phys. 113 no.2 (1987), 299-339.
  24. M. Bordemann, H.-C. Herbig and S. Waldmann, BRST cohomology and phase space reduc- tion in deformation quantization, Comm. Math. Phys. 210 (2000), 107-144.
  25. M. Bordemann, H.-C. Herbig and M. Plaum, A homological approach to singular reduction in defoemation quantization. in: Singularity theory -Dedicated to Jean-Paul Brasselet on His 60th Birthday, Proceedings of the 2005 Marseille Singularity School and Conference, CIRM, Marseille, 2005, edited by Denis Chéniot et. al.
  26. H.-C. Herbig Variations on homological reduction, Dissertation, Goethe-Universität Frank- furt a. M. (2007).
  27. J. Huebschmann and T. Kadeishvili, Small models for chain algebras, Math. Z. 207 (1991), 245-280.
  28. J. P. Brennan, M. V. Pinto, and W. V. Vasconcelos, , The Jacobian module of a Lie algebra., Trans. Amer. Math. Soc. 321 (1990), 183-196.
  29. R. Cushman and R. Sjamaar, On singular reduction of Hamiltonian spaces. in: Symplectic geometry and mathematical physics (Aix-en-Provence, 1990), edited by P. Donato et. al., Progress in Mathematics 99.
  30. J. Arms, M. Gotay and G. Jennings, Geometric and algebraic reduction for singular mo- mentum maps, Adv. Math. 79 (1990), 43-103.
  31. H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, Volume 8, second edition, Cambridge 1989.
  32. E. Bierstone and G. W. Schwarz, Continuous linear division and extension of C ∞ functions, Duke Math. J. 50 (1983), 233-271.
  33. V.L. Ginzburg, Equivariant Poisson cohomology and a spectral sequence associated with a moment map, Internat. J. Math. 10 (1999), 977-1010.
  34. J. Stasheff, Homological reduction of constrained Poisson algebras, J. Differential Geom. 45 (1997), 221-240.
  35. A. Alekssev, P. Xu, Courant algebroids and derived brackets, preprint 2002.
  36. R. Caseiro, Modular classes of Poisson-Nijenhuis Lie algebroids, preprint arXiv:math/0701476.
  37. P. A. Damianou, R. Fernandes, Integrable hierarchies and the modular class, preprint arXiv:math/0607784.
  38. S. Evens, J.-H. Lu, A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Ser.2 50 (1999), 417-436.
  39. J. Grabowski, G. Marmo, P. W. Michor, Homology and modular classes of Lie algebroids, Ann. Inst. Fourier (Grenoble) 56 (2006), 69-83.
  40. Y. Kosmann-Schwarzbach, Modular vector fields and Batalin-Vilkovisky algebras, in Poisson Geometry, J. Grabowski, P. Urbanski, eds., Banach Center Publ. 51 (2000), 109-129.
  41. Y. Kosmann-Schwarzbach, C. Laurent-Gengoux, The modular class of a twisted Poisson structure, Travaux mathématiques (Luxembourg) (2005) 16, 315-339.
  42. Y. Kosmann-Schwarzbach, C. Laurent-Gengoux, A. Weinstein, in preparation.
  43. Y. Kosmann-Schwarzbach, F. Magri, On the modular classes of Poisson-Nijenhuis manifolds, preprint arXiv:math/0611202.
  44. Y. Kosmann-Schwarzbach, A. Weinstein, Relative modular classes of Lie algebroids, C. R. Acad. Sci. Paris, Ser. I 341 (2005), 509-514.
  45. Y. Kosmann-Schwarzbach, M. Yakimov, Modular classes of regular twisted Poisson struc- tures on Lie algebroids, Lett. Math. Phys. 80 (2007), 183-197.
  46. J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in Élie Cartan et les mathématiques d'aujourd'hui, Astérisque (1985), Numéro hors série, 257-271.
  47. E. Meinrenken, Lectures on pure spinors and moment maps, preprint arXiv:math/0609319.
  48. P. Ševera, A. Weinstein, Poisson geometry with a 3-form background, in Noncommutative Geometry and String Theory, Progr. Theoret. Phys. Suppl. no. 144 (2001), 145-154.
  49. A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379-394.
  50. M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), 575-620.
  51. M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Differential Geom. 66 (2004), 71-137.
  52. M. Crainic and R. L. Fernandes and D. Martínez Torres, Poisson manifolds of compact type, in preparation.
  53. D. Kotschick, Free circle actions with contractible orbits on symplectic manifolds, Math. Z. 252 (2006), no. 1, 19-25.
  54. Nguyen Tien Zung, Proper Groupoids and Momentum Maps: Linearization, Affinity and Convexity, math.SG/0407208
  55. S. Bochner, Compact groups of differentiable transformations. Ann. of Math. (2) 46, (1945). 372-381.
  56. M. Chaperon, Quelques outils de la théorie des actions différentiables Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), 259-275, Astérisque, 107-108, Soc. Math. France, Paris, 1983.
  57. C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85-124.
  58. J. Conn, Normal forms for Analytic Poisson structures,Ann. of Math(2)119, n3, 577-601.
  59. J. Conn, Normal forms for smooth Poisson structures. Ann. of Math. (2) 121 (1985), no. 3, 565-593.
  60. V. Ginzburg, Momentum mappings and Poisson cohomology. Internat. J. Math. 7 (1996), no. 3, 329-358.
  61. V. Ginzburg, V. Guillemin and Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions, AMS, 2004.
  62. R. Hamilton, The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65-222.
  63. R. Hamilton, Deformation theory for foliations, Preprint Cornell University.
  64. E. Miranda, Ph. Monnier and Nguyen Tien Zung, Rigidity of Poisson group actions, preprint 2007.
  65. E. Miranda and N. T. Zung, A note on equivariant normal forms of Poisson structures, Math. Research Notes, 2006, vol 13-6, 1001-1012.
  66. R. Palais, Equivalence of nearby differentiable actions of a compact group. Bull. Amer. Math. Soc. 67 1961 362-364.
  67. Y. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf. Lie algebroids and related topics in differential geometry (Warsaw, 2000), 249-274, Banach Center Publ., 54, Polish Acad. Sci., Warsaw, 2001.
  68. A. Weinstein, Lectures on symplectic manifolds. Regional Conference Series in Mathematics, No.29. American Mathematical Society, Providence, R.I., 1977.
  69. A. Weinstein, The local structure of Poisson manifolds., J. Differential Geom. 18 (1983), no. 3, 523-557. References
  70. B. Chen and S. Hu. A deRham model for Chen-Ruan cohomology ring of Abelian orbifolds, Math. Ann. 336, 51-71 (2006).
  71. W. Chen and Y. Ruan: A new cohomology theory of orbifold, Comm. Math. Phys. 248, no. 1, 1-31 (2004).
  72. N. Neumaier, M. Pflaum, H. Posthuma and X. Tang: Homology of of formal deformations of proper étale Lie groupoids. J. Reine u. Angew. Math. 593,
  73. M. Pflaum, H. Posthuma, X.Tang and H.-H. Tseng. Orbifold cup products and ring struc- tures on Hochschild cohomologies. Preprint.
  74. P. Xu and A. Weinstein. Hochschild cohomology and characteristic classes for star-products. In; Geometry of differential equations. Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, RI p. 177-194 1998
  75. References
  76. M.F . Atiyah, Convexity and commuting hamiltonians, Bull. London Math. Soc., 14(1982), pp. 1-15.
  77. P. Birtea, J.-P. Ortega, and T.S. Ratiu, Openness and convexity for momentum maps, Trans. Amer. Math. Soc. 32 (2007), 100-120.
  78. P. Birtea, J.-P. Ortega, and T.S. Ratiu, Metric convexity in the symplectic category, preprint, 2007.
  79. M. P. Condevaux, P. Dazord, and P. Molino, Geometrie du moment, Travaux du Séminaire Sud-Rhodanien de Géométrie, I, Publ. Dép. Math. Nouvelle Sér. B, 88-1, Univ. Claude- Bernard, Lyon, 1988, 131-160.
  80. H. Flaschka, H. and T.S. Ratiu, A convexity theorem for Poisson actions of compact Lie groups, Ann. École Normale Supérieure, 29(1996), 787-811.
  81. V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67(1982), 491-513.
  82. V. Guillemin and S. Sternberg, Convexity properties of the moment mapping II, Invent. Math., 77(1984), 533-546.
  83. J. Hilgert, K.-H. Neeb, and W. Plank, Symplectic convexity theorems and coadjoint ornits, Compositio Math., 94(1994), 129-180.
  84. F.C. Kirwan, Convexity properties of the moment mapping III, Invent. Math., 77(1984), 547-552.
  85. J.-P. Ortega and T.S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics 222, Birkhäuser Boston, 2004.
  86. J.-P. Ortega and T.S. Ratiu, The reduced spaces of a symplectic Lie group action, Ann. Global Anal. Geom., 30(2006), 335-381.
  87. J.-P. Ortega and T.S. Ratiu, The stratified spaces of a symplectic Lie group action, Rep. Math. Phys., 58(1)(2006), 51-75. References
  88. Bursztyn, H., Waldmann, S.: Deformation Quantization of Hermitian Vector Bundles. Lett. Math. Phys. 53 (2000), 349-365.
  89. Waldmann, S.: States and Representation Theory in Deformation Quantization. Rev. Math. Phys. 17 (2005), 15-75.
  90. Weiß, S.: Nichtkommutative Eichtheorien und Deformationsquantisierung von Haupt- faserb"undeln. master thesis, Fakultät für Mathematik und Physik, Physikalisches Institut, Albert-Ludwigs-Universität, Freiburg, 2006. References
  91. A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J., 126 (2005), 1-52.
  92. K. A. Brown, K. A. Goodearl, and M. Yakimov, Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space, Advances in Math. 206 (2006), 567-629.
  93. M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Diff. Geom. 66 (2004), 71-137.
  94. V. V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math., 79 (1985), 499-511.
  95. K. A. Goodearl and M. Yakimov, Poisson structures on affine spaces and flag varieties. II, preprint math.QA/0509075.
  96. G. Lusztig, Total positivity in partial flag manifolds, Representation Theory 2 (1998), 70-78.
  97. R. Richardson, G. Röhrle, and R. Steinberg, Parabolic subgroups with Abelian unipotent radical, Invent. Math. 110 (1992), 649-671.
  98. Ben Webster and Milen Yakimov, A Deodhar type stratification on the double flag variety, preprint, math.SG/0607374, Transformation Groups, in press.
  99. P. Xu, Dirac submanifolds and Poisson involutions, Ann. Sci. École Norm. Sup. (4) 36 (2003), 403-430.
  100. K. Behrend and P. Xu, Differentiable Stacks and Gerbes, arxiv:math.DG/0605694.
  101. L. Breen, Bitorseurs et cohomologie non abélienne, in The Grothendieck Festschrift, Vol. I, volume 86 of Progr. Math., pages 401-476. Birkhäuser Boston, Boston, MA, 1990.
  102. H. Bursztyn and C. Zhu, Morita equivalence of weinstein groupoids, in preparation.
  103. H.-H. Tseng and C. Zhu, Integrating Lie algebroids via stacks, Compos. Math., 142(1):251- 270, 2006.
  104. H.-H. Tseng and C. Zhu, Integrating Poisson manifolds via stacks, Travaux mathématique, 15:285-297, 2006.
  105. A. Weinstein and P. Xu, Extensions of symplectic groupoids and quantization, J. Reine Angew. Math., 417:159-189, 1991.
  106. P. Xu, Morita equivalence of Poisson manifolds, Comm. Math. Phys., 142(3):493-509, 1991.
  107. C. Zhu, Lie n-groupoids and stacky Lie groupoids, arXiv:math.DG/0609420. Reporter: Rui Loja Fernandes