Reductions of Invariant bi-Poisson Structures and Locally Free Actions

Abstract

Let (X,G,ω1,ω2,{ηt}) be a manifold with a bi-Poisson structure {ηt} generated by a pair of G-invariant symplectic structures ω1 and ω2, where a Lie group G acts properly on X. We prove that there exists two canonically defined manifolds (RLi,Gi,ω1i,ω2i,{ηit}), i=1,2 such that (1) RLi is a submanifold of an open dense subset X(H)⊂X; (2) symplectic structures ω1i and ω2i, generating a bi-Poisson structure {ηit}, are Gi- invariant and coincide with restrictions ω1|RLi and ω2|RLi; (3) the canonically defined group Gi acts properly and locally freely on RLi; (4) orbit spaces X(H)/G and RLi/Gi are canonically diffeomorphic smooth manifolds; (5) spaces of G-invariant functions on X(H) and Gi-invariant functions on RLi are isomorphic as Poisson algebras with the bi-Poisson structures {ηt} and {ηit} respectively. The second Poisson algebra of functions can be treated as the reduction of the first one with respect to a locally free action of a symmetry group.