Advertisement

Momentum Mappings And Reduction of Poisson Actions

  • Jiang-Hua Lu
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 20)

Abstract

An action σ: G × PP of a Poisson Lie group G on a Poisson manifold P is called a Poisson action if σ is a Poisson map. It is believed that Poisson actions should be used to understand the “hidden symmetries” of certain integrable systems [STS2]. If the Poisson Lie group G has the zero Poisson structure, then σ being a Poisson action is equivalent to each transformation σ g : PP for gG preserving the Poisson structure on P. In this case, if the orbit space G \ P is a smooth manifold, it has a reduced Poisson structure such that the projection map PG \ P is a Poisson map. If P is symplectic and if the action σ is generated by an equivariant momentum mapping J: Pg*, the reduction procedure of Meyer [Me] and Marsden and Weinstein [Ms-We] gives a way of describing the symplectic leaves of G \ P as the quotients P µ := G µ \J −1 (µ), where µg* and G µ G is the coadjoint isotropy subgroup of µ.

Keywords

Symplectic Manifold Poisson Structure Dual Group Momentum Mapping Poisson Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ab-Ma]
    Abraham, R., Marsden, J.E., “Foundations of mechanics,” 2nd ed., Benjamin/Cummings, New York, 1978.MATHGoogle Scholar
  2. [Bo]
    Bourbaki, N., “Groupes et algèbres de Lie,” Chapitres 2 et 3, Hermann, Paris, 1972.Google Scholar
  3. [Dt-D-W]
    Coste, A., Dazord, P., Weinstein, A., Groupoides symplectiques,(notes d’un cours de A. Weinstein), Publ. Dept. Math. (1987), Université Claude Bernard Lyon I. Google Scholar
  4. [Da-Sol]
    Dazord, P., Sondaz, D., Variétés de Poisson-algébroides de Lie,Séminaire Sud-Rhodanien (1988–1/B), Publ. Univ. Claude Bernard-Lyon 1.Google Scholar
  5. [Da-So2]
    Dazord, P., Sondaz, D.,Groupes de Poisson affines,Proceedings of the Seminaire Sud-Rhodanien de Geometrie (1989) (to appear), Springer-MSRI series.Google Scholar
  6. [Dr]
    Drinfel’d, V.G., Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations, Soviet Math. Dokl. 27 (1) (1983), 68–71.MATHMathSciNetGoogle Scholar
  7. [KS]
    Kosmann-Schwarzbach, Y., Poisson-Drinfel’d groups,Topics in soliton theory and exactly solvable nonlinear equations, M. Ablowitz, B. Fuchsteiner and M. Kruskal, eds. (1987), 191–215, World Scientific, Singapore.Google Scholar
  8. [KS-Ma]
    Kosmann-Schwarzbach, Y., Magri, F., Poisson-Lie groups and complete integrability, part 1, Drinfel’d bialgebras, dual extensions and their canonical representations, Annales Inst. Henri Poincaré, Série A (Physique Théorique) 49 (4) (1988), 433–460.Google Scholar
  9. [Lu-We]
    Lu, J.H., Weinstein, A., Poisson Lie groups, dressing transformations, and Bruhat decompositions, Journal of Differential Geometry 31 (1990), 501–526.MATHMathSciNetGoogle Scholar
  10. [Lu]
    Lu, J.H., Multiplicative and affine Poisson structures on Lie groups, PhD thesis (1990), University of California, Berkeley.Google Scholar
  11. [Ms-We]
    Marsden, J., Weinstein, A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121–129.CrossRefMATHMathSciNetGoogle Scholar
  12. [Me]
    Meyer, K.R., Symmetries and integrals in mechanics, in “Dynamical Systems,” M.N. Peixoto, ed., Academic Press, New York, 1973, pp. 259–272.Google Scholar
  13. [STS1]
    Semenov-Tian-Shansky, M.A., What is a classical r-matrix ?, Funct. Anal. Appl. 17 (4) (1983), 259–272.Google Scholar
  14. [STS2]
    Semenov-Tian-Shansky, M.A., Dressing transformations and Poisson group actions, Publ. RIMS, Kyoto University 21 (1985), 1237–1260.CrossRefMathSciNetGoogle Scholar
  15. [Sp]
    Spivak, M., “A comprehensive introduction to differential geometry,” Vol. 1, Publish or Perish, 1970.Google Scholar
  16. [Wel]
    Weinstein, A., The local structure of Poisson manifolds, J. Diff. Geometry 18 (1983), 523–557.MATHGoogle Scholar
  17. [We2]
    Weinstein, A., Poisson geometry of the principal series and nonlinearizable structures, J. Diff. Geometry 23 (1987), 55–73.Google Scholar
  18. [We3]
    Weinstein, A., Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (4) (1988), 705–727.CrossRefMATHMathSciNetGoogle Scholar
  19. [We4]
    Weinstein, A., Some remarks on dressing transformations, J. Fac. Sci. Univ. Tokyo. Sect. 1A, Math. 36 (1988), 163–167.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1991

Authors and Affiliations

  • Jiang-Hua Lu
    • 1
  1. 1.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

Personalised recommendations