Comptes Rendus
no. 3
Differential Geometry
Equivariant gerbes over compact simple Lie groups
[Gerbes equivariantes sur les groupes de Lie simples compacts]

Présenté par : Charles-Michel Marle

Kai Behrend1 ; Ping Xu2 ; Bin Zhang3
1 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver BC, V6T IZ2, Canada
2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
3 Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794-3600, USA
Comptes Rendus. Mathématique, Volume 336 (2003) no. 3, pp. 251-256.

En utilisant des extensions S1-centrales de groupoı̈des, nous présentons, dans le cas d'un groupe simple compact G, un modèle de dimension infinie d'une S1-gerbe sur un champ différentiable G/G dont la classe de Dixmier–Douady correspond au générateur canonique de la cohomologie équivariante HG3(G).

Using groupoid S1-central extensions, we present, for a compact simple Lie group G, an infinite dimensional model of S1-gerbe over the differential stack G/G whose Dixmier–Douady class corresponds to the canonical generator of the equivariant cohomology HG3(G).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)00024-9
Kai Behrend 1 ; Ping Xu 2 ; Bin Zhang 3

1 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver BC, V6T IZ2, Canada
2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
3 Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794-3600, USA 
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     title = {Equivariant gerbes over compact simple {Lie} groups},
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     pages = {251--256},
     publisher = {Elsevier},
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     year = {2003},
     doi = {10.1016/S1631-073X(02)00024-9},
     language = {en},
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Kai Behrend; Ping Xu; Bin Zhang. Equivariant gerbes over compact simple Lie groups. Comptes Rendus. Mathématique, Volume 336 (2003) no. 3, pp. 251-256. doi : 10.1016/S1631-073X(02)00024-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)00024-9/

[1] A. Alekseev; A. Malkin; E. Meinrenken Lie group valued moment maps, J. Differential Geom., Volume 48 (1998), pp. 445-495

[2] K. Behrend; P. Xu S1-bundles and gerbes over differential stacks, C. R. Acad. Sci. Paris Sér. I, Volume 336 (2003)

[3] K. Behrend, P. Xu, Differential stacks and gerbes, in preparation

[4] J.-L. Brylinski Gerbes on complex reductive Lie groups | arXiv

[5] J. Huebschmann; K. Guruprasad; L. Jeffrey; A. Weinstein Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J., Volume 89 (1997), pp. 377-412

[6] E. Meinrenken The basic gerbe over a compact simple Lie group | arXiv

[7] A. Pressley; G. Segal Loop Groups, Oxford University Press, New York, 1986

[8] A. Weinstein The symplectic structure on moduli space, The Floer Memorial Volume, Progr. Math., 133, 1995, pp. 627-635

[9] A. Weinstein; P. Xu Extensions of symplectic groupoids and quantization, J. Reine Angew. Math., Volume 417 (1991), pp. 159-189

[10] P. Xu Morita equivalent symplectic groupoids (P. Dazord; A. Weinstein, eds.), Symplectic Geometry, Groupoids, and Integrable Systems, Seminaire sud Rhodanien a Berkeley, 1989, 1991, pp. 291-311

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