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We consider higher generalizations of both a (twisted) Poisson structure andthe equivariant condition of a momentum map on a symplectic manifold. On a Liealgebroid over a (pre-)symplectic and (pre-)multisymplectic manifold, weintroduce a Lie algebroid differential form called a compatible E-n-form. Thisdifferential form satisfies a compatibility condition, which is consistent withboth the Lie algebroid structure and the (pre-)(multi)symplectic structure.There are many interesting examples such as a Poisson structure, a twistedPoisson structure and a twisted R-Poisson structure for a pre-n-plecticmanifold. Moreover, momentum maps and momentum sections on symplecticmanifolds, homotopy momentum maps and homotopy momentum sections onmultisymplectic manifolds have this structure.

Compatible E-differential forms on Lie algebroids over (pre-)multisymplectic manifolds