Transformation Groups

, Volume 16, Issue 1, pp 175–191

Induced Dirac structures on isotropy-type manifolds

Article

Abstract

A new method of singular reduction is extended from Poisson to Dirac manifolds. Then it is shown that the Dirac structures on the strata of the quotient coincide with those of the only other known singular Dirac reduction method.

Key words and phrases

Dirac structures singular reduction proper action 

AMS classification

70H45, 70G65, 53C15, 53C10 (primary) 70G45, 53D17, 53D99 (secondary) 

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References

  1. [BC05]
    H. Bursztyn, M. Crainic, Dirac structures, momentum maps, and quasi-Poisson manifolds, in: The Breadth of Symplectic and Poisson Geometry, Progress in Mathematics, Vol. 232, Birkhäuser Boston, Boston, MA, 2005, pp. 1–40, MR MR2103001 (2006e:53147).Google Scholar
  2. [BCG07]
    H. Bursztyn, G. R. Cavalcanti, M. Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007), no. 2, 726–765.CrossRefMATHMathSciNetGoogle Scholar
  3. [Bie75]
    E. Bierstone, Lifting isotopies from orbit spaces, Topology 14 (1975), no. 3, 245–252, MR MR0375356 (51 #11551).CrossRefMATHMathSciNetGoogle Scholar
  4. [Bla00]
    G. Blankenstein, Implicit Hamiltonian systems: Symmetry and interconnection, PhD Thesis, University of Twente, 2000.Google Scholar
  5. [BR04]
    G. Blankenstein, T. S. Ratiu, Singular reduction of implicit Hamiltonian systems, Rep. Math. Phys. 53 (2004), no. 2, 211–260, MR MR2068644 (2005k:37126).CrossRefMATHMathSciNetGoogle Scholar
  6. [BvdS01]
    G. Blankenstein, A. J. van der Schaft, Symmetry and reduction in implicit generalized Hamiltonian systems, Rep. Math. Phys. 47 (2001), no. 1, 57–100, MR MR1823009 (2002e:37083).CrossRefMATHMathSciNetGoogle Scholar
  7. [Cou90]
    T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631–661.CrossRefMATHMathSciNetGoogle Scholar
  8. [CW88]
    T. J. Courant, A. Weinstein, Beyond Poisson structures, in: Action Hamiltoniennes de Groupes, Troisième théorème de Lie (Lyon, 1986), Travaux en Cours, Vol. 27, Hermann, Paris, 1988, pp. 39–49, MR MR951168 (89h:58054).Google Scholar
  9. [DK00]
    J. J. Duistermaat, J. A. C. Kolk, Lie Groups, Universitext, Springer, Berlin, 2000, viii, 344 pp.Google Scholar
  10. [FOR09]
    R. L. Fernandes, J.-P. Ortega, T. S. Ratiu, The momentum map in Poisson geometry, Amer. J. Math. 131 (2009), 1261–1310.CrossRefMATHMathSciNetGoogle Scholar
  11. [JR08]
    M. Jotz, T. S. Ratiu, Dirac and nonholonomic reduction, arXiv:0806.1261v2 (2008).Google Scholar
  12. [JR09]
    M. Jotz, T. S. Ratiu, Poisson reduction by distributions, Lett. Math. Phys. 87 (2009), nos. 1–2, 139–147, MR MR2480650.CrossRefMATHMathSciNetGoogle Scholar
  13. [JR10]
    M. Jotz, T. S. Ratiu, Optimal Dirac reduction, arXiv:1008.2283v1 (2010).Google Scholar
  14. [JRS10]
    M. Jotz, T. S. Ratiu, J. Sniatycki, Singular reduction of Dirac structures, arXiv:0901.3062v5, to appear in Trans. Amer. Math. Soc. (2011).Google Scholar
  15. [JRZ10]
    M. Jotz, T. S. Ratiu, M. Zambon, Invariant frames for vector bundles and applications., preprint (2010).Google Scholar
  16. [LŚ08]
    T. Lusala, J. Śniatycki, Stratified subcartesian spaces, arXiv:0805.4807v1, to appear in Canad. Math. Bull. (2008).Google Scholar
  17. [MR86]
    J. E. Marsden, T. S. Ratiu, Reduction of Poisson manifolds, Lett. Math. Phys. 11 (1986), no. 2, 161–169, MR MR836071 (87h:58067).CrossRefMATHMathSciNetGoogle Scholar
  18. [OR04]
    J.-P. Ortega, T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, Vol. 222, Birkhäuser Boston, Boston, MA, 2004, xxxiv, 497 pp.Google Scholar
  19. [Pfl01]
    M. J. Paum, Analytic and Geometric Study of Stratified Spaces, Lecture Notes in Mathematics, Vol. 1768, Springer-Verlag, Berlin, 2001, MR MR1869601 (2002m:58007).Google Scholar
  20. [Śni03]
    J. Śniatycki, Orbits of families of vector fields on subcartesian spaces, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 7, 2257–2296, MR MR2044173 (2005k:53160).Google Scholar
  21. [SX08]
    M. Stiénon, P. Xu, Reduction of generalized complex structures, J. Geom. Phys. 58 (2008), no. 1, 105–121, MR 2378459 (2008k:53184).CrossRefMATHMathSciNetGoogle Scholar
  22. [YM07]
    H. Yoshimura, J. E. Marsden, Reduction of Dirac structures and the Hamilton–Pontryagin principle, Rep. Math. Phys. 60 (2007), no. 3, 381–426, MR MR2369539 (2009m:37157).CrossRefMATHMathSciNetGoogle Scholar
  23. [YM09]
    H. Yoshimura, J. E. Marsden, Dirac cotangent bundle reduction, J. Geom. Mech. 1 (2009), no. 1, 87–158, MR MR2511303.CrossRefMATHMathSciNetGoogle Scholar
  24. [Zam08]
    M. Zambon, Reduction of branes in generalized complex geometry, J. Symplectic Geom. 6 (2008), no. 4, 353–378, MR MR2471097.MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Section de MathématiquesEcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Section de Mathématiques and Bernouilli CenterEcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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