= Fixed-point normalization removes the moment-map cocycle
{title2=$\{\mu_\xi,\mu_\eta\}=\mu_{[\xi,\eta]}$}
Suppose a symplectic <Lie group action> on a connected <symplectic manifold> has global component Hamiltonians and a common fixed point $p_0$. Choose their constants linearly so $\mu_\xi(p_0)=0$. The difference $c(\xi,\eta)=\{\mu_\xi,\mu_\eta\}-\mu_{[\xi,\eta]}$ has zero <Hamiltonian vector field> because fundamental fields and Hamiltonian fields have compatible Lie-bracket conventions; it is therefore constant. At $p_0$, all fundamental fields vanish, so both terms in $c$ vanish. Thus $c=0$ everywhere and the <Poisson brackets> of the components reproduce the group <Lie algebra>. For a connected group this also yields coadjoint equivariance. A linear unitary action on a vector space has the origin as such a fixed point.
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