Symplectic Geometry, Groupoids, and Integrable Systems

Volume 20 of the series Mathematical Sciences Research Institute Publications pp 209-226

Momentum Mappings And Reduction of Poisson Actions

  • Jiang-Hua LuAffiliated withDepartment of Mathematics, University of California at Berkeley

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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.