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 associated to any Lie algebroid . 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.
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 associée à une algébroïde de Lie . 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.
Keywords: Lie algebroids, classifying spaces, equivariant cohomology
Mot clés : algebroide de Lie, espaces classifiants, cohomologie équivariant
Arias Abad, Camilo 1; Crainic, Marius 2
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TY - JOUR AU - Arias Abad, Camilo AU - Crainic, Marius TI - The Weil algebra and the Van Est isomorphism JO - Annales de l'Institut Fourier PY - 2011 SP - 927 EP - 970 VL - 61 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2633/ DO - 10.5802/aif.2633 LA - en ID - AIF_2011__61_3_927_0 ER -
%0 Journal Article %A Arias Abad, Camilo %A Crainic, Marius %T The Weil algebra and the Van Est isomorphism %J Annales de l'Institut Fourier %D 2011 %P 927-970 %V 61 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2633/ %R 10.5802/aif.2633 %G en %F AIF_2011__61_3_927_0
Arias Abad, Camilo; Crainic, Marius. The Weil algebra and the Van Est isomorphism. Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 927-970. doi : 10.5802/aif.2633. https://aif.centre-mersenne.org/articles/10.5802/aif.2633/
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