
SIGMA 11 (2015), 058, 10 pages arXiv:1502.06253
https://doi.org/10.3842/SIGMA.2015.058
Contribution to the Special Issue on Poisson Geometry in Mathematics and Physics
Modular Classes of Lie Groupoid Representations up to Homotopy
Rajan Amit Mehta
Department of Mathematics & Statistics, Smith College, 44 College Lane, Northampton, MA 01063, USA
Received February 24, 2015, in final form July 23, 2015; Published online July 25, 2015
Abstract
We describe a construction of the modular class associated to a representation up to homotopy of a Lie groupoid. In the case of the adjoint representation up to homotopy, this class is the obstruction to the existence of a volume form, in the sense of Weinstein's ''The volume of a differentiable stack''.
Key words:
Lie groupoid; representation up to homotopy; modular class.
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References

Arias Abad C., Crainic M., Representations up to homotopy and Bott's spectral sequence for Lie groupoids, Adv. Math. 248 (2013), 416452, arXiv:0911.2859.

Crainic M., Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), 681721, math.DG/0008064.

Crainic M., Fernandes R.L., Secondary characteristic classes of Lie algebroids, in Quantum Field Theory and Noncommutative Geometry, Lecture Notes in Phys., Vol. 662, Springer, Berlin, 2005, 157176.

del Hoyo M.L., Fernandes R.L., Riemannian metrics on Lie groupoids, arXiv:1404.5989.

Evens S., Lu J.H., Weinstein A., Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. (2) 50 (1999), 417436, dgga/9610008.

Fernandes R.L., Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), 119179, math.DG/0007132.

GraciaSaz A., Mehta R.A., Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Adv. Math. 223 (2010), 12361275, arXiv:0810.0066.

GraciaSaz A., Mehta R.A., VBgroupoids and representation theory of Lie groupoids, arXiv:1007.3658.

KosmannSchwarzbach Y., Poisson manifolds, Lie algebroids, modular classes: a survey, SIGMA 4 (2008), 005, 30 pages, arXiv:0710.3098.

Mehta R.A., Supergroupoids, double structures, and equivariant cohomology, Ph.D. Thesis, University of California, Berkeley, 2006, math.DG/0605356.

Mehta R.A., Lie algebroid modules and representations up to homotopy, Indag. Math. (N.S.) 25 (2014), 11221134, arXiv:1107.1539.

Weinstein A., The volume of a differentiable stack, Lett. Math. Phys. 90 (2009), 353371, arXiv:0809.2130.

Weinstein A., Xu P., Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), 159189.

Yamagami S., Modular cohomology class of foliation and Takesaki's duality, in Geometric Methods in Operator Algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser., Vol. 123, Longman Sci. Tech., Harlow, 1986, 415439.

