On Poisson actions of compact Lie groups on symplectic manifolds

Abstract

Let $G_{\P}$ be a compact simple Poisson-Lie group equipped with a Poisson structure $\P$ and $(M, \o)$ be a symplectic manifold. Assume that $M$ carries a Poisson action of $G_{\P}$ and there is an equivariant moment map in the sense of Lu and Weinstein which acts to the dual Poisson-Lie group $G^*_{\P}$, $\m: M\rightarrow G^*_{\P}$. We prove that $M$ always possesses another symplectic form $\to$ so that the $G$-action preserves $\tilde{\o}$ and there is a new moment map $\mu= e^{-1} \circ \m: M\rightarrow \g^*$. Here $e$ is a universal (independent of $M$) invertible equivariant map $e: \g^*\rightarrow G^*_{\P}$. We suggest new short proves of the convexity theorem for the Poisson-Lie moment map, Poisson reduction theorem and the Ginzburg-Weinstein theorem on the isomorphism of $\g^*$ and $G^*_{\P}$ as Poisson spaces.