The Weil algebra and the Van Est isomorphism

Details The Weil algebra and the Van Est isomorphism The Weil algebra and the Van Est isomorphism

2009-01-03

arxiv.org/abs/0901.0322v2

Mathematics

Differential Geometry

28 pages. Final version, to appear in "Annales de l'Institut Fourier"

Scientific paper

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.

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