On poisson actions of compact lie groups on symplectic manifolds

Abstract

Let Gp be a compact simple Poisson-Lie group equipped with a Poisson structured, and (M,ω) be a symplectic manifold. Assume that M carries a Poisson action of Gp, and there is an equivariant moment map in the sense of Lu and Weinstein which maps M to the dual Poisson-Lie group Gp, m : M → Gp. We prove that M always possesses another symplectic form tli so that the G-action preserves UJ and there is a new moment map μ = e-1 o m : M → G*. Here e is a universal (independent of M) invertible equivariant map e : G* → Gp. We suggest new short proofs of the convexity theorem for the Poisson-Lie moment map, the Poisson reduction theorem and the Ginzburg-Weinstein t heorem on the isomorphism of �* and Gp as Poisson spaces. © 1997 J. differential geometry.