  We consider higher generalizations of both a (twisted) Poisson structure and
the equivariant condition of a momentum map on a symplectic manifold. On a Lie
algebroid over a (pre-)symplectic and (pre-)multisymplectic manifold, we
introduce a Lie algebroid differential form called a compatible $E$-$n$-form.
This differential form satisfies a compatibility condition, which is consistent
with both the Lie algebroid structure and the (pre-)(multi)symplectic
structure. There are many interesting examples such as a Poisson structure, a
twisted Poisson structure and a twisted $R$-Poisson structure for a
pre-$n$-plectic manifold. Moreover, momentum maps and momentum sections on
symplectic manifolds, homotopy momentum maps and homotopy momentum sections on
multisymplectic manifolds have this structure.
