The Weil algebra and the Van Est isomorphism

Camilo Arias Abad[1]; Marius Crainic[2]

  • [1] Universität Zürich Institut für Mathematik Zürich (Switzerland)
  • [2] Utrecht University Department of Mathematics Utrecht (The Netherlands)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 3, page 927-970
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Arias Abad, Camilo, and Crainic, Marius. "The Weil algebra and the Van Est isomorphism." Annales de l’institut Fourier 61.3 (2011): 927-970. <http://eudml.org/doc/219714>.

@article{AriasAbad2011,
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.},
affiliation = {Universität Zürich Institut für Mathematik Zürich (Switzerland); Utrecht University Department of Mathematics Utrecht (The Netherlands)},
author = {Arias Abad, Camilo, Crainic, Marius},
journal = {Annales de l’institut Fourier},
keywords = {Lie algebroids; classifying spaces; equivariant cohomology; Weil algebra; Van Est isomorphism},
language = {eng},
number = {3},
pages = {927-970},
publisher = {Association des Annales de l’institut Fourier},
title = {The Weil algebra and the Van Est isomorphism},
url = {http://eudml.org/doc/219714},
volume = {61},
year = {2011},
}

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
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 3
SP - 927
EP - 970
AB - 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.
LA - eng
KW - Lie algebroids; classifying spaces; equivariant cohomology; Weil algebra; Van Est isomorphism
UR - http://eudml.org/doc/219714
ER -

References

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  17. R. Mehta, Supergroupoids, double structures and equivariant cohomology, (2006) MR2709144
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