Duke Mathematical Journal

Poisson homogeneous spaces and Lie algebroids associated to Poisson actions

Jiang-Hua Lu

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Article information

Duke Math. J. Volume 86, Number 2 (1997), 261-304.

First available in Project Euclid: 19 February 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58H05: Pseudogroups and differentiable groupoids [See also 22A22, 22E65]
Secondary: 22E60: Lie algebras of Lie groups {For the algebraic theory of Lie algebras, see 17Bxx} 57S25: Groups acting on specific manifolds 58F05


Lu, Jiang-Hua. Poisson homogeneous spaces and Lie algebroids associated to Poisson actions. Duke Math. J. 86 (1997), no. 2, 261--304. doi:10.1215/S0012-7094-97-08608-7. http://projecteuclid.org/euclid.dmj/1077242667.

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  • [Am] R. Aminou, Bigèbres de Lie et groupes de Lie-Poisson, thesis, Université de Lille, 1988.
  • [BB] A. Beilinson and J. Bernstein, A proof of Jantzen conjectures, I. M. Gelfand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, 1993, pp. 1–50.
  • [CDW] 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, i–ii, 1–62.
  • [DaSo] P. Dazord and D. Sondaz, Groupes de Poisson affines, Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, California, 1989), Math. Sci. Res. Inst. Publ., vol. 20, Springer-Verlag, New York, 1991, pp. 99–128.
  • [Dr1] V. G. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations, Soviet Math. Dokl. 27 (1983), 68–71.
  • [Dr2] V. G. Drinfeld, On Poisson homogeneous spaces of Poisson-Lie groups, Theoret. and Math. Phys. 95 (1993), 524–525.
  • [HM] P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra 129 (1990), no. 1, 194–230.
  • [Ka] E. Karolinsky, Symplectic leaves on Poisson homogeneous spaces of Poisson Lie groups, preprint.
  • [KS] Y. Kosmann-Schwarzbach, Poisson-Drinfeld groups, Topics in Soliton Theory and Exactly Solvable Nonlinear Equations (Oberwolfach, 1986), World Scientific, Singapore, 1987, pp. 191–215.
  • [Kt1] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387.
  • [Kt2] B. Kostant, Lie algebra cohomology and generalized Schubert cells, Ann. of Math. (2) 77 (1963), 72–144.
  • [Ku] J. L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque (1985), no. numero hors série, 257–271, Mathematical Heritage of Elie Cartan, Lyon, 1984.
  • [Li] A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom. 12 (1977), no. 2, 253–300.
  • [Lu1] J. H. Lu, Multiplicative and affine Poisson structures on Lie groups, Ph.D. thesis, University of California, Berkeley, 1990.
  • [Lu2] J. H. Lu, Momentum mappings and reduction of Poisson actions, Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, California 1989), Math. Sci. Res. Inst. Publ., vol. 20, Springer-Verlag, New York, 1991, pp. 209–226.
  • [LuWe] J. H. Lu and A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501–526.
  • [Mcz1] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Math. Soc. Lecture Note Ser., vol. 124, Cambridge University Press, Cambridge, 1987.
  • [Mcz2] K. Mackenzie, Double Lie algebroids and second-order geometry I, Adv. Math. 94 (1992), no. 2, 180–239.
  • [MX] K. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (1994), no. 2, 415–452.
  • [Mj] S. Majid, Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations, Pacific J. Math. 141 (1990), no. 2, 311–332.
  • [Mk] T. Mokri, Matched pairs of Lie algebroids, to appear in Glasgow Math. J., 1997.
  • [STS] M. A. Semenov-Tian-Shansky, Dressing transformations and Poisson group actions, Publ. Res. Inst. Math. Sci. 21 (1985), no. 6, 1237–1260.
  • [We] A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), no. 3, 523–557.
  • [Xu] P. Xu, On Poisson groupoids, Internat. J. Math. 6 (1995), no. 1, 101–124.
  • [Za] S. Zakrzewski, Poisson homogeneous spaces, Quantum Groups, Formalism and Applications, Proceedings of the XXX Winter School of Theoretical Physics, 14–26 January 1994, Karpacz eds. J. Lukierski, Z. Popwicz, and J. Sobczyk, Polish Scientific Publishers PWN, Warsaw, 1995, pp. 629–639.