Momentum Mappings And Reduction of Poisson Actions

Abstract

An action σ: G × P→P 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 : P→ P for g ∈ G 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 P→G \ P is a Poisson map. If P is symplectic and if the action σ is generated by an equivariant momentum mapping J: P→ g*, 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 µ∈ g* and G _µ ⊂ G is the coadjoint isotropy subgroup of µ.