Let $G$ be a connected finite-dimensional Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, and let $(M,\theta)$ be a connected symplectic manifold. Suppose $G$ acts smoothly on $M$ on the left by symplectomorphisms, and suppose the action is Hamiltonian with moment map
Moreover $\sigma$ is a Chevalley-Eilenberg $2$-cocycle for $\mathfrak g$ with real coefficients carrying the zero $\mathfrak g$-action. With the convention
for $\ell\in\mathfrak g^*$, the moment map can be made $G$-equivariant by replacing $\mu$ with $\mu+a$ for some constant $a\in\mathfrak g^*$ if and only if $\sigma$ is a Chevalley-Eilenberg coboundary.