Modules and representations up to homotopy of Lie $n$-algebroids

This paper studies differential graded modules and representations up to homotopy of Lie $n$-algebroids, for general $n\in\mathbb{N}$. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie $n$-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie $n$-algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie $n$-algebroids are used to encode decomposed VB-Lie $n$-algebroid structures on double vector bundles.