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, 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(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 : 10.5802/aif.2633
Classification : 58H05, 53D17, 55R40
Keywords: Lie algebroids, classifying spaces, equivariant cohomology
Mot clés : algebroide de Lie, espaces classifiants, cohomologie équivariant
Arias Abad, Camilo 1 ; Crainic, Marius 2

1 Universität Zürich Institut für Mathematik Zürich (Switzerland)
2 Utrecht University Department of Mathematics Utrecht (The Netherlands)
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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/articles/10.5802/aif.2633/

[1] Arias Abad, C.; Crainic, M. Representations up to homotopy and Bott’s spectral sequence for Lie groupoids (preprint arXiv:0911.2859, submitted for publication)

[2] Arias Abad, C.; Crainic, M. Representations up to homotopy of Lie algebroids (preprint arXiv:0901.0319, submitted for publication)

[3] Berline, Nicole; Getzler, Ezra; Vergne, Michèle Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004 (Corrected reprint of the 1992 original) | MR | Zbl

[4] Bott, R. On the Chern-Weil homomorphism and the continuous cohomology of Lie-groups, Advances in Math., Volume 11 (1973), pp. 289-303 | DOI | MR | Zbl

[5] Bott, R.; Shulman, H.; Stasheff, J. On the de Rham theory of certain classifying spaces, Advances in Math., Volume 20 (1976) no. 1, pp. 43-56 | DOI | MR | Zbl

[6] Bursztyn, Henrique; Crainic, Marius; Weinstein, Alan; Zhu, Chenchang Integration of twisted Dirac brackets, Duke Math. J., Volume 123 (2004) no. 3, pp. 549-607 | DOI | MR | Zbl

[7] Cartan, Henri Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie, Colloque de Topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, 1951, pp. 15-27 | Zbl

[8] Crainic, Marius Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv., Volume 78 (2003) no. 4, pp. 681-721 | DOI | MR | Zbl

[9] Crainic, Marius Prequantization and Lie brackets, J. Symplectic Geom., Volume 2 (2004) no. 4, pp. 579-602 http://projecteuclid.org/getRecord?id=euclid.jsg/1144070630 | MR | Zbl

[10] Crainic, Marius; Fernandes, Rui Loja Integrability of Lie brackets, Ann. of Math. (2), Volume 157 (2003) no. 2, pp. 575-620 | DOI | MR | Zbl

[11] van Est, W. T. Group cohomology and Lie algebra cohomology in Lie groups. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math., Volume 15 (1953), p. 484-492, 493–504 | MR | Zbl

[12] Guillemin, Victor W.; Sternberg, Shlomo Supersymmetry and equivariant de Rham theory, Mathematics Past and Present, Springer-Verlag, Berlin, 1999 With an appendix containing two reprints by Henri Cartan [MR0042426 (13,107e); MR0042427 (13,107f)] | MR | Zbl

[13] Haefliger, André Groupoïdes d’holonomie et classifiants, Astérisque (1984) no. 116, pp. 70-97 Transversal structure of foliations (Toulouse, 1982) | Numdam | Zbl

[14] Kamber, Franz W.; Tondeur, Philippe Foliated bundles and characteristic classes, Lecture Notes in Mathematics, Vol. 493, Springer-Verlag, Berlin, 1975 | MR | Zbl

[15] Mackenzie, Kirill C. H. General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005 | MR | Zbl

[16] Mathai, Varghese; Quillen, Daniel Superconnections, Thom classes, and equivariant differential forms, Topology, Volume 25 (1986) no. 1, pp. 85-110 | DOI | MR | Zbl

[17] Mehta, R. Supergroupoids, double structures and equivariant cohomology, Berkeley (2006) (Ph. D. Thesis Arxiv math/0605356) | MR

[18] Moerdijk, I.; Mrčun, J. Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003 | MR | Zbl

[19] Segal, Graeme Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. (1968) no. 34, pp. 105-112 | DOI | EuDML | Numdam | MR | Zbl

[20] Weinstein, Alan Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.), Volume 16 (1987) no. 1, pp. 101-104 | DOI | MR | Zbl

[21] Weinstein, Alan; Xu, Ping Extensions of symplectic groupoids and quantization, J. Reine Angew. Math., Volume 417 (1991), pp. 159-189 | EuDML | MR | Zbl

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