The Weil algebra and the Van Est isomorphism

TL;DR

This paper introduces a generalized Weil algebra for Lie algebroids, establishes a Van Est isomorphism relating it to classifying space cohomology, and applies this to simplify and extend results in Poisson geometry.

Contribution

It generalizes the Weil algebra to Lie algebroids and proves a Van Est isomorphism, connecting classifying space cohomology with Lie algebroid structures.

Findings

Established a Van Est isomorphism for the Weil algebra of Lie algebroids.

Simplified proofs of reconstruction results for multiplicative forms.

Linked Weil algebra and Van Est maps to Poisson and Dirac manifold quantization.

Abstract

This paper belongs to a series devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman's BRST model, here we introduce the Weil algebra $W(A)$ associated to any Lie algebroid $A$. We then show that this Weil algebra is related to the Bott-Shulman-Stasheff complex (computing the cohomology of the classifying space) via a Van Est map and we prove a Van Est isomorphism theorem. As application, we generalize and find a simpler more conceptual proof of the main result of Bursztyn et.al. on the reconstructions of multiplicative forms and of a result of Weinstein-Xu and Crainic on the reconstruction of connection 1-forms. This reveals the relevance of the Weil algebra and Van Est maps to the integration and the pre-quantization of Poisson (and Dirac) manifolds.