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

Abstract

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.

Typos and signs corrected, relation to other work, references and details in computations have been added, distinction between left and right modules have been made