Weil Algebras and Double Lie Algebroids

Given a double vector bundle D -> M, we define a bigraded bundle of algebras W(D) -> M called the Weil algebra bundle of D. The space W(D) of sections of this algebra realizes the algebra of functions on the supermanifold D[1,1]. We describe in detail the relations between the Weil algebra bundles of D and those of the double vector bundles D', D'' obtained from D by duality operations. We show that VB algebroid structures on D are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gerstenhaber bracket on the third. Furthermore, Mackenzie's definition of a double Lie algebroid is equivalent to compatibilities between two such structures on any one of the three Weil algebras. In particular, we obtain a classical version of Voronov's result characterizing double Lie algebroid structures. In the case that D=TA is the tangent prolongation of a Lie algebroid, we find that W(D) is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad-Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multivector fields, and 2-term representations up to homotopy all have natural interpretations in terms of our Weil algebras.Ph.D