The Weil algebra and the Van Est isomorphism

Arias Abad, Camilo; Crainic, Marius

Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 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 $W\left(A\right)$ 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 $W\left(A\right)$ 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.

Classification : 58H05, 53D17, 55R40
Keywords: Lie algebroids, classifying spaces, equivariant cohomology
Mot clés : algebroide de Lie, espaces classifiants, cohomologie équivariant

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