The Weil algebra of a double Lie algebroid

Abstract

Given a double vector bundle $D\\to M$, we define a bigraded `Weil algebra $\\mathcal{W}(D)$, which `realizes the algebra of smooth functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebras of $D$ and those of the double vector bundles $D ,\\ D $ obtained by duality operations. In particular, we show that double-linear Poisson structures on $D$ can be described alternatively as Gerstenhaber brackets on $\\mathcal{W}(D)$, vertical differentials on $\\mathcal{W}(D )$, or horizontal differentials on $\\mathcal{W}(D )$. We also give a new proof 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 $\\mathcal{W}(D)$ 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.