On Poisson actions of compact Lie groups on symplectic manifolds

Let $G_{\mathrm{\P }}$ be a compact simple Poisson-Lie group equipped with a Poisson structure $\mathrm{\P }$ and $(M,\text{o})$ be a symplectic manifold. Assume that $M$ carries a Poisson action of $G_{\mathrm{\P }}$ and there is an equivariant moment map in the sense of Lu and Weinstein which acts to the dual Poisson-Lie group $G_{\mathrm{\P }}^{\ast }$, $\text{m}:M\to G_{\mathrm{\P }}^{\ast }$. We prove that $M$ always possesses another symplectic form $\to$ so that the $G$-action preserves $\tilde{\text{o}}$ and there is a new moment map $\mu =e^{-1}\circ \text{m}:M\to {\text{g}}^{\ast }$. 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 ${\text{g}}^{\ast }$ and $G_{\mathrm{\P }}^{\ast }$ as Poisson spaces.