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.
Highlights
Two Poisson structures, η1 and η2, are said to be compatible if the sum η1 + η2, or, equivalently, any linear combination ηt = t1η1 + t2η2, t = (t1, t2) ∈ R2 is a Poisson structure
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
Summary
The submanifold (X, G) is very special, allowing to canonically identify the spaces AG of G-invariant functions on X and AG of G-invariant functions on X and the bi-Poisson structure {(ηt) } induced on AG AG can be treated as the reduction with respect to a locally free action of a Lie group. Given a proper action of a connected Lie group G on a connected manifold X and an isotropy subgroup H ⊂ G representing the principal orbit type, consider the subset X(H) of X consisting of the points in X with the stabilizer conjugated to H in G. We illustrate the theory by a class of examples of reductions of bi-Poisson structures on cotangent bundles to coadjoint orbits (homogeneous spaces) G/K, where a compact Lie group G acts on G/K and on the cotangent bundle T∗(G/K) by the lifted action (see Section 3). To fix the notation we shall prove this fact below
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