# The Weil algebra of a double Lie algebroid

## TL;DR

This paper introduces a bigraded Weil algebra for double Lie algebroids, linking it to various structures like Poisson brackets, Lie algebroid deformations, and representations up to homotopy, providing new insights and proofs.

## Contribution

It defines a new Weil algebra for double Lie algebroids, explores its relations with duality and Poisson structures, and connects it to existing theories like deformation complexes and IM forms.

## Key findings

- Weil algebra $\\mathcal{W}(D)$ realizes functions on $D[1,1]$.
- Double-linear Poisson structures correspond to Gerstenhaber brackets and differentials on Weil algebras.
- Special case: $D=TA$ recovers the Weil algebra of a 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.