= Moment-map equivariance obstruction
{title2=$\kappa(X,Y)=\{\mu_X,\mu_Y\}-\mu_{[X,Y]}$}
For a weakly <Hamiltonian> left <Lie group action> on a connected <symplectic manifold>, choose $\iota_{X_f}\omega=df$. <Fundamental vector fields> and <Hamiltonian vector fields> are both antihomomorphisms, so the displayed difference is constant. The <Jacobi identity for the Poisson bracket> makes it a two-cocycle. Changing $\mu_X$ by $b(X)$ changes the difference by $-b([X,Y])$. Thus its class in <Lie algebra cohomology> obstructs choosing an equivariant <moment map>. For a connected group, vanishing of this class gives infinitesimal and group equivariance.
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