Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 025, 19 pages      arXiv:2302.08193      https://doi.org/10.3842/SIGMA.2024.025

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

Noriaki Ikeda
Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan

Received November 13, 2023, in final form March 27, 2024; Published online March 31, 2024

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.

Key words: Poisson geometry; Lie algebroid; multisymplectic geometry; higher structures.

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