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Lie-Poisson structure on some Poisson Lie groups


Authors: Viktor L. Ginzburg and Alan Weinstein
Journal: J. Amer. Math. Soc. 5 (1992), 445-453
MSC: Primary 17B56; Secondary 22E60, 58F05
DOI: https://doi.org/10.1090/S0894-0347-1992-1126117-8
MathSciNet review: 1126117
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  • [A] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15. MR 642416 (83e:53037)
  • [ABo] M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1-28. MR 721448 (85e:58041)
  • [B] G. Bredon, Introduction to compact transformation groups, Pure and Appl. Math. vol. 46, Academic Press, New York, 1972. MR 0413144 (54:1265)
  • [C] J. Conn, Normal forms for smooth Poisson structures, Ann. of Math. (2) 121 (1985), 565-593. MR 794374 (86m:58050)
  • [D] T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339. MR 984900 (90b:58069)
  • [Dr1] V. G. Drinfel'd, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations, Soviet Math. Dokl. 27 (1983), 68-71. MR 688240 (84i:58044)
  • [Dr2] -, Quantum groups, Proc. ICM, Berkeley 1 (1986), 789-820.
  • [Du] J. J. Duistermaat, On the similarity between the Iwasawa projection and the diagonal part, Société Mathématique de France, 2e série, Mémoire no. 15, 1984, pp. 129-138. MR 789082 (86k:22035)
  • [E] E. T. van Est, Une application d'une méthode de Cartan-Leary, Indag. Math. 17 (1955), 542-544. MR 0073108 (17:385a)
  • [F] D. B. Fuks, Cohomology of infinite-dimensional Lie algebras, Consultants Bureau, New York, 1986. MR 874337 (88b:17001)
  • [GL] V. L. Ginzburg and J. -H. Lu, Poisson calculus: Vector fields and differential forms (in preparation).
  • [GS] V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491-513. MR 664117 (83m:58037)
  • [H] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, 1978. MR 514561 (80k:53081)
  • [K] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Math. Notes, vol. 31, Princeton Univ. Press, Princeton, NJ, 1984. MR 766741 (86i:58050)
  • [Ko] B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Sup. (4) 6 (1973), 413-455. MR 0364552 (51:806)
  • [Ku] J. L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Élie Cartan et les Mathématiques d'Aujourd'hui, Astérisque hors série (1985), 257-271. MR 837203 (88m:17013)
  • [L] J.-H. Lu, Multiplicative and affine Poisson structures on Lie groups, Berkeley Thesis, 1990.
  • [LR] J.-H. Lu and T. Ratiu, On the nonlinear convexity theorem of Kostant, J. Amer. Math. Soc. 4 (1991), 349-363. MR 1086967 (92a:58048)
  • [LW] J.-H. Lu and A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), 501-526. MR 1037412 (91c:22012)
  • [M] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 280-296. MR 0182927 (32:409)
  • [W] A. Weinstein, Some remarks on dressing transformations, J. Fac. Sci. Univ. Tokyo, Sect. A, Math. 36 (1988), 163-167. MR 931446 (89j:58036)
  • [WX] A. Weinstein and P. Xu, Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), 159-189. MR 1103911 (92k:58094)

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DOI: https://doi.org/10.1090/S0894-0347-1992-1126117-8
Article copyright: © Copyright 1992 American Mathematical Society

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