Existence and unicity of co-moments in multisymplectic geometry

Given a multisymplectic manifold $(M,\omega)$ and a Lie algebra $\frak{g}$ acting on it by infinitesimal symmetries, Fregier-Rogers-Zambon define a homotopy (co-)moment as an $L_{\infty}$-algebra-homomorphism from $\frak{g}$ to the observable algebra $L(M,\omega)$ associated to $(M,\omega)$, in analogy with and generalizing the notion of a co-moment map in symplectic geometry. We give a cohomological characterization of existence and unicity for homotopy co-moment maps and show its utility in multisymplectic geometry by applying it to special cases as exact multisymplectic manifolds and simple Lie groups and by deriving from it existence results concerning partial co-moment maps, as e.g. covariant multimomentum maps and multi-moment maps.