# The Weil algebra and the Van Est isomorphism

Abad, C A; Crainic, M (2011). The Weil algebra and the Van Est isomorphism. Annales de l'institut Fourier, 61(3):927-970.

## Abstract

This paper belongs to a series of papers 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 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 [6] on the reconstructions of multiplicative forms and of a result of [21, 9] 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.

## Abstract

This paper belongs to a series of papers 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 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 [6] on the reconstructions of multiplicative forms and of a result of [21, 9] 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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Other titles: Algèbre de Weil et isomorphisme de Van Est Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Algebra and Number Theory Physical Sciences > Geometry and Topology English 2011 17 Feb 2012 20:18 11 Mar 2020 22:51 Association des Annales de l'Institut Fourier 0373-0956 (P) 1777-5310 (E) Green https://doi.org/10.5802/aif.2633 http://arxiv.org/abs/0901.0322

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