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 [1], Lie algebroid modules [2] and representations up to homotopy of Lie algebroids [3]. 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 [4], 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 [5] and offer a wide generalisation of the notion of a VB-groupoid. We show that a refined version of the Van Est theorem [6] holds for weighted Lie groupoids, and in fact follows from minor adjustments to the ideas and proofs presented by Cabrera & Drummond [7].

References
[1] Bruce A.J., Grabowska K., Grabowski J., Linear duals of graded bundles and higher analogues of (Lie) algebroids, J. Geom. Phys. 101 (2016), 71–99.

[2] Vaintrob A.Yu., Lie algebroids and homological vector fields, Russ. Math. Surv. 52 (1997), 428–429.

[3] Abad C.A., Crainic M., Representations up to homotopy of Lie algebroids, J. Reine Angew.Math, 663 (2012), 91–126.

[4] Mehta R.A., Lie algebroid modules and representations up to homotopy. Indag. Math. (N.S.) 25 (2014), no. 5, 1122–1134.

[5] Bruce A.J., Grabowska K., Grabowski J., Graded Bundles in the Category of Lie Groupoids, SIGMA 11 (2015), 090, 25 pages.

[6] Crainic M., Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv, 78 (2003), 681–72.

[7] Cabrera A., Drummond T., Van Est isomorphism for homogeneous cochains, Pacific J. Math. 287 (2017), 297–336