Representations theory of Lie algebroids and weighted Lie algebroids

board 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].

[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