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Momentum Mappings And Reduction of Poisson Actions

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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

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© 1991 Springer-Verlag New York, Inc.

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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. https://doi.org/10.1007/978-1-4613-9719-9_15

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  • DOI: https://doi.org/10.1007/978-1-4613-9719-9_15

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