Skip to main content
SpringerLink
Log in

REDUCTION OF SYMPLECTIC GROUPOIDS AND QUOTIENTS OF QUASI-POISSON MANIFOLDS

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

In this work, we study the integrability of quotients of quasi-Poisson manifolds. Our approach allows us to put several classical results about the integrability of Poisson quotients in a common framework. By categorifying one of the already known methods of reducing symplectic groupoids we also describe double symplectic groupoids, which integrate the recently introduced Poisson groupoid structures on gauge groupoids.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Álvarez, Integrability of Quotients in Poisson and Dirac Geometry, PhD thesis, IMPA, Rio de Janeiro, 2019.

  5. Álvarez, D.: Integrability of quotients in Poisson and Dirac geometry. Pacif. J. Math. 311(1), 1–32 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Álvarez, D.: Leaves of stacky Lie algebroids. Compt. Rend. Math. 358(2), 217–226 (2020)

    MathSciNet  MATH  Google Scholar 

  7. Behrend, K., Xu, P., Zhang, B.: Equivariant gerbes over compact simple Lie groups. Compt. Rend. Math. 336(3), 251–256 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brown, R., Mackenzie, K.C.H.: Determination of a double Lie groupoid by its core diagram. J. Pure Appl. Algebra. 80(3), 237–272 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bursztyn, H., Cabrera, A.: Multiplicative forms at the infinitesimal level. Math. Ann. 353, 663–705 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bursztyn, H., Cabrera, A., del Hoyo, M.: Vector bundles over Lie groupoids and algebroids. Adv. Math. 290, 163–207 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Cabrera, C. Ortiz, Quotients of multiplicative forms and Poisson reduction, arXiv:2008.03416 (2020).

  13. A. Cattaneo, G. Felder, Poisson sigma models and symplectic groupoids in: Quantization of Singular Symplectic Quotients, Progr. Math., Vol 198, Birkhäuser, Basel, 2001, pp. 61–93.

  14. Chen, Z., Stiénon, M., Xu, P.: Poisson 2-groups. J. Diff. Geom. 94(2), 209–240 (2013)

    MATH  Google Scholar 

  15. Crainic, M.: Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes. Comm. Math. Helvetici. 78, 681–721 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Crainic, M., Fernandes, R.L.: A geometric approach to Conns linearization theorem. Ann. of Math. 173(2), 1121–1139 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Crainic, M., Fernandes, R.L.: Integrability of Lie brackets. Ann. of Math. 157(2), 575–620 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Crainic, M., Fernandes, R.L.: Integrability of Poisson brackets. J. Diff. Geom. 66(1), 71–137 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Crainic, M., Fernandes, R.L., Martínez, D.: Poisson manifolds of compact types (PMCT 1). J. Reine Ang. Math. 756, 101–149 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. В. Г. Дринфельд, Гамильтоновы структуры на группах Ли, биалгебры Ли и геометрический смысл классических уравнений Янга–Бакстера, Докл. АН СССР 268 (1983), ном. 2, 285–287. Engl. transl.: V. G. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang–Baxter equations, Soviet Math. Dokl. 27 (1983), 68–71.

  21. J. P. Dufour, N. T. Zung, Poisson Structures and Their Normal Forms, Progr. Math., Vol. 242, Birkhäuser Verlag, Basel, 2006.

  22. Fernandes, R.L., Ortega, J.P., Ratiu, T.S.: The moment map in Poisson geometry. Amer. J. Math. 131(5), 1261–1310 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. R. L. Fernandes, D. Iglesias Ponte, Integrability of Poisson–Lie group actions, Letters Math. Phys. 90 (2009), no. 1, 137–159.

  24. Hawkins, E.: A groupoid approach to quantization. J. Symplectic Geom. 6(1), 61–125 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Higgins, P.J., Mackenzie, K.C.H.: Algebraic constructions inthe category of Lie algebroids. J. Algebra. 129(1), 194–230 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. М. В. Карасёв, Аналоги объектов теории групп Ли для нелинейных скобок Пуассона, Изв. АН СССР. Сер. матем 50 (1986), вьш. 3, 508–538. Eng. transl.: M. V. Karasev, Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, Math. USSR Izvestiya 28 (1987), 497–527.

  27. D. Li-Bland, P. Severa, Quasi-Hamiltonian groupoids and multiplicative Manin pairs, Internat. Math. Res. Not. 2011 (2011), no. 10, 2295–2350.

  28. J.-H. Lu, Multiplicative and Affine Poisson Structures on Lie Groups, PhD thesis, University of California, Berkeley, 1990.

  29. J.-H. Lu, Some examples of algebraic symplectic groupoids (seminar talk), in: Global Poisson Webspace, 2020.

  30. Lu, J.-H., Mouquin, V.: Double Bruhat cells and symplectic groupoids. Transform. Groups. 23, 765–800 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lu, J.-H., Mouquin, V.: Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras. Internat. Math. Res. Not. 2017(19), 5919–5976 (2017)

    MathSciNet  MATH  Google Scholar 

  32. J.-H. Lu, A. Weinstein, Groupoïdes symplectiques doubles des groupes de Lie–Poisson, C. R. Acad. Sci. Paris 309 (1989), no. 18, 951–954.

  33. Mackenzie, K.C.H.: Double Lie algebroids and second-order geometry, I. Adv. Math. 94(2), 180–239 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mackenzie, K.C.H.: Double Lie algebroids and second-order geometry, II. Adv. Math. 154(1), 46–75 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc. Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.

  36. Mackenzie, K.C.H., Xu, P.: Integration of Lie bialgebroids. Topology. 39(3), 445–467 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mackenzie, K.C.H., Xu, P.: Lie bialgebroids and Poisson groupoids. Duke Math. J. 73(2), 415–452 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mikami, K., Weinstein, A.: Moments and reduction for symplectic groupoids. Publ. Res. Inst. Math. Sci. 24(1), 121–140 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  39. I. Moerdijk, J. Mrcun, Introduction to Foliations and Lie Groupoids, Cambridge Studies in Adv. Math.,Vol. 91, Cambridge University Press, Cambridge, 2003.

  40. Moerdijk, I., Mrcun, J.: On integrability of infinitesimal actions. Amer. J. Math. 124(3), 567–593 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  41. Iglesias Ponte, D., Laurent-Gengoux, C., Xu, P.: Universal lifting theorem and quasi-Poisson groupoids. J. Europ. Math. Soc. 14(3), 681–731 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. D. Roytenberg, Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds, PhD thesis, University of California, Berkeley, 1999.

  43. Semenov-Tian-Shansky, M.A.: Dressing transformations and Poisson group actions. Publ. RIMS, Kyoto Univ. 21(6), 1237–1260 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  44. L. Stefanini, On Morphic Actions and Integrability of LA-groupoids, PhD thesis, ETH Zürich, 2008.

  45. Vaintrob, A.Y.: Lie algebroids and homological vector fields. Russian Math. Surveys. 52(2), 428–429 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  46. Weinstein, A.: Coisotropic calculus and Poisson groupoids. J. Math. Soc. Japan. 40(4), 705–727 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  47. Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. Amer. Math. Soc. 16(1), 101–104 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Facultad de Ciencias UNAM, CP 04510, Mexico City, México

    DANIEL ÁLVAREZ

Authors
  1. DANIEL ÁLVAREZ
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to DANIEL ÁLVAREZ.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

ÁLVAREZ, D. REDUCTION OF SYMPLECTIC GROUPOIDS AND QUOTIENTS OF QUASI-POISSON MANIFOLDS. Transformation Groups 28, 1357–1374 (2023). https://doi.org/10.1007/s00031-022-09700-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-022-09700-4

Search

Navigation