Equivariant gerbes over compact simple Lie groups

Kai Behrend; Ping Xu; Bin Zhang

doi:10.1016/S1631-073X(02)00024-9

Differential Geometry

Equivariant gerbes over compact simple Lie groups
[Gerbes equivariantes sur les groupes de Lie simples compacts]

Présenté par : Charles-Michel Marle

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

Comptes Rendus. Mathématique, Volume 336 (2003) no. 3, pp. 251-256.

Abstract (VO)
Résumé

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 H_G³(G).

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 H_G³(G).

Reçu le : 2002-09-24
Accepté le : 2002-12-02
Publié le : 2003-02-01

DOI : 10.1016/S1631-073X(02)00024-9

Affiliations des auteurs :

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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     author = {Kai Behrend and Ping Xu and Bin Zhang},
     title = {Equivariant gerbes over compact simple {Lie} groups},
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     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/

Bibliographie
Cité par

[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

Daniel Álvarez Reduction of symplectic groupoids and quotients of quasi-Poisson manifolds, Transformation Groups, Volume 28 (2023) no. 4, pp. 1357-1374 | DOI:10.1007/s00031-022-09700-4 | Zbl:1531.53085
Byungdo Park; Corbett Redden A classification of equivariant gerbe connections, Communications in Contemporary Mathematics, Volume 21 (2019) no. 2, p. 40 (Id/No 1850001) | DOI:10.1142/s0219199718500013 | Zbl:1442.53020
Derek Krepski Groupoid equivariant prequantization, Communications in Mathematical Physics, Volume 360 (2018) no. 1, pp. 169-195 | DOI:10.1007/s00220-017-3080-x | Zbl:1454.22001
Derek Krepski Basic equivariant gerbes on non-simply connected compact simple Lie groups, Journal of Geometry and Physics, Volume 133 (2018), pp. 30-41 | DOI:10.1016/j.geomphys.2018.06.016 | Zbl:1398.53031
Chi-Kwong Fok Picard group of isotropic realizations of twisted Poisson manifolds, Journal of Geometric Mechanics, Volume 8 (2016) no. 2, pp. 179-197 | DOI:10.3934/jgm.2016003 | Zbl:1352.53068
D. Li-Bland; P. Severa Quasi-Hamiltonian Groupoids and Multiplicative Manin Pairs, International Mathematics Research Notices (2010) | DOI:10.1093/imrn/rnq170
Camille Laurent-Gengoux; Mathieu Stiénon; Ping Xu Non-abelian differentiable gerbes, Advances in Mathematics, Volume 220 (2009) no. 5, pp. 1357-1427 | DOI:10.1016/j.aim.2008.10.018 | Zbl:1177.22001
Jean-Louis Tu; Ping Xu The ring structure for equivariant twisted K-theory, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2009 (2009) no. 635 | DOI:10.1515/crelle.2009.077
Michael Murray; Danny Stevenson The basic bundle gerbe on unitary groups, Journal of Geometry and Physics, Volume 58 (2008) no. 11, pp. 1571-1590 | DOI:10.1016/j.geomphys.2008.07.006 | Zbl:1154.55011
Henrique Bursztyn; Marius Crainic Dirac structures, momentum maps, and quasi-Poisson manifolds, The Breadth of Symplectic and Poisson Geometry, Volume 232 (2007), p. 1 | DOI:10.1007/0-8176-4419-9_1
Camille Laurent-Gengoux; Ping Xu Quantization of pre-quasi-symplectic groupoids and their Hamiltonian spaces, The Breadth of Symplectic and Poisson Geometry, Volume 232 (2007), p. 423 | DOI:10.1007/0-8176-4419-9_14
Z. Shahbazi Prequantization of Quasi-Hamiltonian spaces, International Mathematics Research Notices (2006) | DOI:10.1155/imrn/2006/29354
A. Cabrera; H. Montani Hamiltonian loop group actions and T-duality for group manifolds, Journal of Geometry and Physics, Volume 56 (2006) no. 7, p. 1116 | DOI:10.1016/j.geomphys.2005.06.006
Krzysztof Gawedzki Abelian and non-abelian branes in WZW models and gerbes, Communications in Mathematical Physics, Volume 258 (2005) no. 1, pp. 23-73 | DOI:10.1007/s00220-005-1301-1 | Zbl:1094.81047
H. Bursztyn On Gauge Transformations of Poisson Structures, Quantum Field Theory and Noncommutative Geometry, Volume 662 (2005), p. 89 | DOI:10.1007/11342786_5
Henrique Bursztyn; Marius Crainic; Alan Weinstein; Chenchang Zhu Integration of twisted Dirac brackets, Duke Mathematical Journal, Volume 123 (2004) no. 3, pp. 549-607 | DOI:10.1215/s0012-7094-04-12335-8 | Zbl:1067.58016
Krzysztof Gawȩdzki; Nuno Reis Basic gerbe over non-simply connected compact groups, Journal of Geometry and Physics, Volume 50 (2004) no. 1-4, pp. 28-55 | DOI:10.1016/j.geomphys.2003.11.004 | Zbl:1067.22009
Eckhard Meinrenken The basic gerbe over a compact simple Lie group, L'Enseignement Mathématique. 2e Série, Volume 49 (2003) no. 3-4, pp. 307-333 | Zbl:1061.53034

Cité par 18 documents. Sources : Crossref, zbMATH

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