Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds

TL;DR

This paper introduces compatible $E$-$n$-forms on Lie algebroids over (pre-)multisymplectic manifolds, unifying various geometric structures like Poisson and momentum maps within a generalized framework.

Contribution

It defines a new type of differential form on Lie algebroids that captures compatibility with both Lie algebroid and multisymplectic structures, extending existing geometric concepts.

Findings

Includes examples such as Poisson, twisted Poisson, and twisted $R$-Poisson structures.

Unifies momentum maps and sections in symplectic and multisymplectic contexts.

Provides a framework for higher generalizations of classical structures.

Abstract

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.