Momentum Mappings And Reduction of Poisson Actions

  • Jiang-Hua Lu
Conference paper

DOI: 10.1007/978-1-4613-9719-9_15

Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 20)
Cite this paper as:
Lu JH. (1991) Momentum Mappings And Reduction of Poisson Actions. In: Dazord P., Weinstein A. (eds) Symplectic Geometry, Groupoids, and Integrable Systems. Mathematical Sciences Research Institute Publications, vol 20. Springer, New York, NY

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 σg: 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 Pµ := Gµ \J−1(µ), where µg* and GµG is the coadjoint isotropy subgroup of µ.

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Copyright information

© Springer-Verlag New York, Inc. 1991

Authors and Affiliations

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

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