Skip to main content
SpringerLink
Log in

Dirac structures, momentum maps, and quasi-Poisson manifolds

  • Chapter
Book cover The Breadth of Symplectic and Poisson Geometry

Part of the book series: Progress in Mathematics ((PM,volume 232))

Abstract

We extend the correspondence between Poisson maps and actions of symplectic groupoids, which generalizes the one between momentum maps and Hamiltonian actions, to the realm of Dirac geometry. As an example, we show how Hamiltonian quasi-Poisson manifolds fit into this framework by constructing an “inversion” procedure relating quasi-Poisson bivectors to twisted Dirac structures.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Alekseev and Y. Kosmann-Schwarzbach, Manin pairs and moment maps, J. Differential Geom., 56 (2000), 133–165.

    MATH  MathSciNet  Google Scholar 

  2. A. Alekseev, Y. Kosmann-Schwarzbach, and E. Meinrenken, Quasi-Poisson manifolds, Canadian J. Math., 54 (2002), 3–29.

    MATH  MathSciNet  Google Scholar 

  3. A. Alekseev, A. Malkin, and E. Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998), 445–495.

    MATH  MathSciNet  Google Scholar 

  4. M. Bangoura and Y. Kosmann-Schwarzbach, The double of a Jacobian quasi-bialgebra, Lett. Math. Phys., 28 (1993), 13–29.

    Article  MATH  MathSciNet  Google Scholar 

  5. K. Behrend, P. Xu, and B. Zhang, Equivariant gerbes over compact simple Lie groups, C. R. Acad. Sci. Paris, 336 (2003), 251–256.

    MATH  MathSciNet  Google Scholar 

  6. H. Bursztyn, M. Crainic, A. Weinstein, and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J., 123-3 (2004), 549–607.

    MathSciNet  Google Scholar 

  7. H. Bursztyn and O. Radko, O, Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier (Grenoble), 53 (2003), 309–337.

    MATH  MathSciNet  Google Scholar 

  8. A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, American Mathematical Society, Providence, RI, 1999.

    MATH  Google Scholar 

  9. A. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients, Progress in Mathematics, Vol. 198, Birkhäuser, Basel, 2001, 61–93.

    Google Scholar 

  10. A. Cattaneo and P. Xu, Integration of twisted Poisson structures, J. Geom. Phys., 49-2 (2004), 187–196.

    Article  MathSciNet  Google Scholar 

  11. A. Coste, P. Dazord, and A. Weinstein, Groupoïdes symplectiques, in Publications du Département de Mathématiques. Nouvelle Série A, Vol. 2, i–ii, Université Claude-Bernard, Lyon, 1987, 1–62.

    Google Scholar 

  12. T. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631–661.

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Courant and A. Weinstein, Beyond Poisson Structures, Séminaire sud-rhodanien de géométrie VIII, Travaux en Cours, Vol. 27, Hermann, Paris, 1988, 39–49.

    Google Scholar 

  14. M. Crainic and R. Fernandes, Integrability of Lie brackets, Ann. Math., 157 (2003), 575–620.

    MATH  MathSciNet  Google Scholar 

  15. M. Crainic and R. Fernandes, Integrability of Poisson Brackets, Math.DG/0210152.

    Google Scholar 

  16. P. Dazord, Groupoïde d’holonomie et géométrie globale, C. R. Acad. Sci. Paris, 324 (1997), 77–80.

    MATH  MathSciNet  Google Scholar 

  17. K. Guruprasad, J. Huebschmann, L. Jeffrey, and A. Weinstein, Groups systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J., 89 (1997), 377–412.

    Article  MATH  MathSciNet  Google Scholar 

  18. P. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra, 129 (1990), 194–230.

    Article  MATH  MathSciNet  Google Scholar 

  19. C. Klimčik and T. Ströbl, WZW-Poisson manifolds, J. Geom. Physics, 4 (2002), 341–344.

    Article  Google Scholar 

  20. J.-H. Lu, Multiplicative and Affine Poisson Structures on Lie Groups, Ph.D. thesis, University of California at Berkeley, Berkeley, 1990.

    Google Scholar 

  21. J.-H. Lu, Momentum mappings and reduction of Poisson actions, in Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Springer-Verlag, New York, 1991, 291–311.

    Google Scholar 

  22. J.-H. Lu and A. Weinstein, Groupoïdes symplectiques double des groupes de Lie-Poisson, C. R. Acad. Sci. Paris, 309 (1989), 951–954.

    MATH  MathSciNet  Google Scholar 

  23. K. Mackenzie and P. Xu, Integration of Lie bialgebroids, Topology, 39 (2000), 445–467.

    Article  MATH  MathSciNet  Google Scholar 

  24. J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys., 5 (1974), 121–130.

    Article  MATH  MathSciNet  Google Scholar 

  25. K. Mikami and A. Weinstein, Moments and reduction for symplectic groupoid actions, Publ. RIMS Kyoto Univ., 24 (1988), 121–140.

    Article  MATH  MathSciNet  Google Scholar 

  26. I. Moerdijk and J. Mrčun., On integrability of infinitesimal actions, Amer. J. Math., 124 (2002), 567–593.

    Article  MATH  MathSciNet  Google Scholar 

  27. J.-S. Park, Topological open p-branes, in Symplectic geometry and Mirror symmetry (Seoul 2000), World Scientific Publishing, River Edge, NJ, 2001, 311–384.

    Google Scholar 

  28. P. Ševera and A. Weinstein, Poisson geometry with a 3-form background, Progr. Theoret. Phys. Suppl., 144 (2001), 145–154.

    Google Scholar 

  29. A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.), 16 (1987), 101–104.

    Article  MATH  MathSciNet  Google Scholar 

  30. A. Weinstein, Poisson geometry. Symplectic geometry, Differential Geom. Appl., 9 (1998), 213–238.

    Article  MathSciNet  Google Scholar 

  31. A. Weinstein, The geometry of momentum, in Proceedings of the Conference on “Geometry in the 20th Century: 1930–2000”, to appear; Math.SG/0208108.

    Google Scholar 

  32. P. Xu, Morita equivalent symplectic groupoids, in Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Springer-Verlag, New York, 1991, 291–311.

    Google Scholar 

  33. P. Xu, Morita Equivalence and Momentum Maps, preprint; Math.SG/0307319.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 3G3, Canada

    Henrique Bursztyn

  2. Department of Mathematics, Utrecht University, P. O. Box 80.010, 3508 TA, Utrecht, The Netherlands

    Marius Crainic

Authors
  1. Henrique Bursztyn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marius Crainic
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Engineering and Applied Science Control and Dynamical Systems, California Institute of Technology, Pasadena, CA, 91125, USA

    Jerrold E. Marsden

  2. Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland

    Tudor S. Ratiu

Additional information

Dedicated to Alan Weinstein for his 60th birthday.

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Boston

About this chapter

Cite this chapter

Bursztyn, H., Crainic, M. (2005). Dirac structures, momentum maps, and quasi-Poisson manifolds. In: Marsden, J.E., Ratiu, T.S. (eds) The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics, vol 232. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4419-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation