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I want to study about Symplectic group actions and moment map, especially Hamiltonian Group Actions. Can you help me with some concretely example of Hamiltonian group actions ? Where can I find some exemple?

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I think that if you say what motivated you to learn about Hamiltonian group actions, the answers you'll receive will be specifically suited to your tastes. In any case, you can always look in McDuff-Salamon's symplectic topology book or the notes Claudio provided a link for. –  Russell Mar 14 '13 at 4:28
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up vote 2 down vote accepted

I think these http://www.math.ist.utl.pt/~acannas/Books/lsg.pdf are very good notes, and freely available.

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Consider the representation of $U(1)$ on $\mathbb{C}^n$ defined by $$t\cdot (z_1,\ldots,z_n)=(tz_1,\ldots,tz_n),$$ where $t\in U(1)$. This action is Hamiltonian.

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Furthermore, a moment map is $\mu:\mathbb{C}^n\rightarrow\frak{u}(1)^*\cong\mathbb{R}$, $$\mu(z)=\frac{1}{2}<z,z>,$$ where <,> denotes the standard Hermitian inner product on $\mathbb{C}^n$. –  Peter Crooks Mar 13 '13 at 23:11
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