|Weighted Lie algebroids are Lie algebroids in the category of graded bundles, or vice versa. It is well known that VB- algebroids (vector bundles in the category of Lie algebroids, or vice versa) are related to 2-term representations up to homotopy of Lie algebroids. Thus, it is natural to wonder if a similar relation holds for weighted Lie algebroids as these are a wide generalization fo VB-algebroids.|
In a preprint entitled “Graded differential geometry and the representation theory of Lie algebroids” with Janusz Grabowski and Luca Vitagliano, we look at the relation between weighted Lie algebroids , Lie algebroid modules  and representations up to homotopy of Lie algebroids . We show that associated with any weighted Lie algebroid is a series of canonical Lie algebroid modules over the underlying weight zero Lie algebroid. Moreover, we know, due to Mehta , that a Lie algebroid module is (up to isomorphisms classes) equivalent to a representation up to homotopy of the Lie algebroid.
Weighted Lie groupoids were first defined and studied in  and offer a wide generalisation of the notion of a VB-groupoid. We show that a refined version of the Van Est theorem  holds for weighted Lie groupoids, and in fact follows from minor adjustments to the ideas and proofs presented by Cabrera & Drummond .
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