Advertisement

The reduced spaces of a symplectic Lie group action

  • Juan-Pablo OrtegaEmail author
  • Tudor S. Ratiu
Article

Abstract

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux et al. In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.

Keywords

Symplectic reduction Marsden–Weintein reduction Poisson reduction Momentum map Lie group action 

References

  1. 1.
    Alekseev, A., Malkin, A. and Meinrenken, E.: Lie group valued momentum maps, J. Differential Geom. 48 (1998), 445–495.zbMATHMathSciNetGoogle Scholar
  2. 2.
    Arms, J. M., Cushman, R. and Gotay, M.J.: A universal reduction procedure for Hamiltonian group actions. In: T.S. Ratiu (ed.), The Geometry of Hamiltonian Systems, Springer Verlag, 1991, p. 33–51.Google Scholar
  3. 3.
    Bates, L. and Lerman, E.: Proper group actions and symplectic stratified spaces, Pacific J. Math. 181(2) (1997), 201–229.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Camacho, C. and Lins Neto, A.: Geometric Theory of Foliations. Birkhäuser, 1985.Google Scholar
  5. 5.
    Cartan, É.: Leçons sur les Invariants Intégraux. Hermann, 1922.Google Scholar
  6. 6.
    Condevaux, M., Dazord, P. and Molino, P.: Géométrie 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.Google Scholar
  7. 7.
    Cushman, R. and Sniatycki, J.: Differential structure of orbit spaces, Canad. J. Math. 53(4) (2001), 715–755.MathSciNetGoogle Scholar
  8. 8.
    Dazord, P.: Feuilletages à singularités. Nederl Akad. Wetensch. Indag. Math. 47 (1985), 21–39.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ginzburg, V.: Some remarks on symplectic actions of compact groups, Math. Z. 210(4) (1992), 625–640.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Guieu, L. and Roger, C.: L'Algébre et le Groupe de Virasoro: Aspects Géométriques et Algébriques, Généralisations. Preprint, 2004.Google Scholar
  11. 11.
    Huebschmann, J.: Symplectic and poisson structures of certain moduli spaces II. Projective representations of cocompact planar discrete groups, Duke Math. J. 80 (1995), 757–770.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Huebschmann, J. and Jeffrey, L.: Group cohomology construction of symplectic forms on certain moduli spaces, Int. Math. Research Notices 6 (1994), 245–249.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry. Volume I. Interscience Tracts in Pure Appl. Math., number 15. John Wiley & Sons, 1963.Google Scholar
  14. 14.
    Marsden, J. E. and Ratiu, T. S.: Reduction of poisson manifolds, Lett. Math. Phys. 11 (1986), 161–169.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Marsden, J. E. and Weinstein, A.: Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5(1) (1974), 121–130.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    McDuff, D.: The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988), 149–160.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ortega, J.-P.: Symmetry, reduction, and stability in hamiltonian systems, Ph.D. Thesis. University of California, Santa Cruz. June, 1998.Google Scholar
  18. 18.
    Ortega, J.-P.: The symplectic reduced spaces of a poisson action, C. R. Acad. Sci. Paris Sér. I Math. 334 (2002), 999–1004.zbMATHMathSciNetGoogle Scholar
  19. 19.
    Ortega, J.-P.: Singular dual pairs, Differential Geom. Appl. 19(1) (2003), 61–95.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ortega, J.-P. and Ratiu, T. S.: The optimal momentum map, in Geometry, Mechanics, and Dynamics. Volume in Honor of the 60th Birthday of J. E. Marsden, P. Newton, Ph. Holmes, A. Weinstein, editors, Springer-Verlag (2002), 329–362.Google Scholar
  21. 21.
    Ortega, J.-P. and Ratiu, T. S.: A symplectic slice theorem. Lett. Math. Phys. 59 (2002a), 81–93.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ortega, J.-P. and Ratiu, T. S.: Momentum Maps and Hamiltonian Reduction. Prog. Math., 222. Birkhäuser-Verlag, Boston, 2004.Google Scholar
  23. 23.
    Ortega, J.-P. and Ratiu, T. S.: The universal covering and covered spaces of a symplectic Lie algebra action, in: J. E. Marsden and T. S. Ratiu, (ed.), The Breadth of Symplectic and Poisson Geometry: Festschrift in Honor of Alan Weinstein, Progress in Mathematics 232, Birkhäuser-Verlag, Boston (2004a), 571–581.Google Scholar
  24. 24.
    Ortega, J.-P. and Ratiu, T. S.: The stratified spaces of a symplectic Lie group action, Rep. Math Phys., to appear (2006)Google Scholar
  25. 25.
    Scheerer, U. and Wulff, C.: Reduced dynamics for momentum maps with cocycles, C. R. Acad. Sci. Paris Sér. I Math. 333(11) (2001), 999–1004.zbMATHMathSciNetGoogle Scholar
  26. 26.
    Sjamaar, R. and Lerman, E.: Stratified symplectic spaces and reduction, Ann. of Math. 134 (1991), 375–422.CrossRefMathSciNetGoogle Scholar
  27. 27.
    Souriau, J.-M.: Structure des systèmes dynamiques. Dunod. Paris, English translation by R. H. Cushman and G. M. Tuynman as Structure of Dynamical Systems. A Symplectic View of Physics, Prog. Math. 149, 1969, Birkhäuser-Verlag, 1997.Google Scholar
  28. 28.
    Stefan, P.: Accessibility and foliations with singularities, Bull. Amer. Math. Soc. 80 (1974), 1142–1145.zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Stefan, P.: Accessible sets, orbits and foliations with singularities, Proc. London Math. Soc. 29 (1974), 699–713.Google Scholar
  30. 30.
    Sussmann, H.: Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Centre National de la Recherche Scientifique, Département de Mathématiques de Besançon, UFR des Sciences et TechniquesUniversité de Franche-ComtéBesançon cedexFrance
  2. 2.Section de Mathématiques and Centre BernoulliÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

Personalised recommendations