We establish a relationship between two different generalizations of Lie algebroid representations: representation up to homotopy and Vaĭntrob’s Lie algebroid modules. Specifically, we show that there is a noncanonical way to obtain a representation up to homotopy from a given Lie algebroid module, and that any two representations up to homotopy obtained in this way are equivalent in a natural sense. We therefore obtain a one-to-one correspondence, up to equivalence.
Lie algebroid, Representation up to homotopy, Graded manifold, Graded vector bundle, Q-manifold
Mehta, Rajan Amit, "Lie algebroid modules and representations up to homotopy" (2014). Mathematics and Statistics: Faculty Publications. 15.