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 $(M,ø)$ 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 $G_{\P }^{\ast }$, $\m: M\rightarrow G^*_¶$. We prove that $M$ always possesses another symplectic form $\to$ so that the $G$-action preserves $\widetilde{ø}$ and there is a new moment map $μ= e^{-1} \circ \m: M\rightarrow \g^*$. Here $e$ is a universal (independent of $M$) invertible equivariant map $e: \g^*\rightarrow G^*_¶$. 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 $G_{\P }^{\ast }$ as Poisson spaces.

arXiv commentLaTeX file, 16 pages

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