Skip to main content Accessibility help
×
Home

The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces

  • David Martínez Torres (a1)

Abstract

A surface $\sum$ endowed with a Poisson tensor $\pi$ is known to admit a canonical integration, $G\left( \pi \right)$ , which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if $\text{ }\!\!\pi\!\!\text{ }$ is not an area form on the 2-sphere, then $G\left( \pi \right)$ is diffeomorphic to the cotangent bundle $T*\sum$ . This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces
      Available formats
      ×

Copyright

References

Hide All
[1] Cattaneo, A. S. and Felder, G., Poisson sigma models and symplecticgroupoids.In: Quantization of singularsymplectic quotients, Progr. Math., 198, Birkhâuser, Basel, 2001, pp. 6193.
[2] Bonechi, F., Ciccoli, N., Staffolani, N., and Tarlini, M., The quantization of the symplecticgroupoid of the standard Podles sphere. J. Geom. Phys. 62(2012), no. 8,18511865. http://dx.doi.Org/1 0.101 6/j.geomphys.2012.04.001
[3] Crainic, M. and Fernandes, R. L., Integrability of Poisson brackets. J. Differential Geom. 66(2004), no. 1, 71137.
[4] Crainic, M. and Màrcuf, I., On the existence of symplectic realizations. J. Symplectic Geom. 9(2011), no. 4, 435444. http://dx.doi.org/10.4310/JSC.2011 .v9.n4.a2
[5] Hawkins, E., A groupoid approach to quantization. J. Symplectic Geom. 6(2008), no. 1, 61125. http://dx.doi.org/10.4310/JSC.2008.v6.n1.a4
[6] Martinez Torres, D., A note on the separability of canonical integrations of Lie algebroids. Math. Res. Lett. 17(2010), no. 1, 6975. http://dx.doi.org/10.4310/MRL.2010.v17.n1 .a6
[7] McDuff, D., The symplectic structure ofKàhlermanifolds of nonpositive curvature. J. Differential Geom. 28(1988), no. 3, 467475.
[8] McDuff, D. and Salamon, D., Introduction to symplectic topology. Second éd.,Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.
[9] Meigniez, G., Submersions, fibrations and bundles. Trans. Amer. Math. Soc. 354(2002), no. 9, 37713787. http://dx.doi.org/10.1090/S0002-9947-02-02972-0
[10] Weinstein, A., Symplecticgroupoids and Poisson manifolds. Bull. Amer. Math.Soc. (N.S.) 16(1987), no. 1, 101104. http://dx.doi.org/10.1090/S0273-0979-1987-15473-5
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Related content

Powered by UNSILO

The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces

  • David Martínez Torres (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.