A Lie algebroid is a vector bundle together with a Lie bracket in the space of sections, and satisfying a Leibnitz rule. There is a standard notion of Lie algebroid representation, which is known to be insufficient, because it does not include an adjoint representation. As it is often the case, going from mere manifolds to the super-world solves this problem.
A VB-algebroid can be defined as a "Lie algebroid object in the category of vector bundles". It turns out that VB-algebroids are the intrinsic objects that correspond to superrepresentations concentrated in two degrees. In other words, a VB-algebroid is to a superrepresentation what a linear map is to a matrix.
I will define all these concepts and explain the relations between them. This is joint work with Rajan A. Mehta.