The Weil algebra and the Van Est isomorphism  [ Algèbre de Weil et isomorphisme de Van Est ]
Annales de l'Institut Fourier, Tome 61 (2011) no. 3, p. 927-970
Cet article fait partie d’ une série consacrée à l’étude de la cohomologie des espaces classifiants. En généralisant l’algèbre de Weil d’une algèbre de Lie et le modèle BRST de Kalkman, nous introduisons l’algèbre de Weil W(A) associée à une algébroïde de Lie A. Nous montrons ensuite que cette algèbre de Weil est liée au complexe de Bott-Shulman (calculant la cohomologie de l’espace classifiant) via une application de Van Est et nous prouvons un théorème d’isomorphisme de type Van Est. Une application de ces méthodes conduit à généraliser de façon plus conceptuelle des reconstitutions de formes multiplicatives et de 1-formes de connexion.
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.
DOI : https://doi.org/10.5802/aif.2633
Classification:  58H05,  53D17,  55R40
Mots clés: algebroide de Lie, espaces classifiants, cohomologie équivariant
@article{AIF_2011__61_3_927_0,
     author = {Arias Abad, Camilo and Crainic, Marius},
     title = {The Weil algebra  and the Van Est isomorphism},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {3},
     year = {2011},
     pages = {927-970},
     doi = {10.5802/aif.2633},
     mrnumber = {2918722},
     zbl = {1237.58021},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_3_927_0}
}
Arias Abad, Camilo; Crainic, Marius. The Weil algebra  and the Van Est isomorphism. Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 927-970. doi : 10.5802/aif.2633. http://www.numdam.org/item/AIF_2011__61_3_927_0/

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