The Weil algebra and the Van Est isomorphism (2011)

Title The Weil algebra and the Van Est isomorphism
Published in Annales de l'Institut Fourier, Vol. 61, No. 3, p.927-. ISSN 0373-0956.
Author Arias Abad, C; Crainic, M.
Date 2011
Language English
Type Article
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 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.
Persistent Identifier URN:NBN:NL:UI:10-1874-234172
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Repository Utrecht University

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