\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

The leaf space of a multiplicative foliation

Abstract Related Papers Cited by
  • We show that if a smooth multiplicative subbundle $S\subseteq TG$ on a groupoid $G⇉P$ is involutive and satisfies completeness conditions, then its leaf space $G/S$ inherits a groupoid structure over the space of leaves of $TP\cap S$ in $P$.
        As an application, a special class of Dirac groupoids is shown to project by forward Dirac maps to Poisson groupoids.
    Mathematics Subject Classification: Primary: 58H05, 53C12; Secondary: 22A22, 53D17.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    A. Coste, P. Dazord and A. Weinstein, Groupoï des symplectiques, Publications du Département de Mathématiques. Nouvelle Série. A, 2, Publ. Dép. Math. Nouvelle Sér. A, 87, Univ. Claude-Bernard, Lyon, 1987, pp. i-ii, 1-62.

    [2]

    T. J. Courant, Dirac manifolds, Trans. Am. Math. Soc., 319 (1990), 631-661.doi: 10.1090/S0002-9947-1990-0998124-1.

    [3]

    J. Hilgert and K.-H. Neeb, Lie Groups and Lie Algebras. (Lie-Gruppen und Lie-Algebren.), Braunschweig: Vieweg. 361 S., 1991.

    [4]

    B. Z. Iliev, "Handbook of Normal Frames and Coordinates," Progress in Mathematical Physics 42. Basel: Birkhäuser. xvi+441 pp., 2006.

    [5]

    D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids, J. Nonlinear Sci., 18 (2008), 221-276. (English).doi: 10.1007/s00332-007-9012-8.

    [6]

    M. Jotz and C. Ortiz, Foliated groupoids and their infinitesimal data, Preprint, arXiv:1109.4515v1. (2011).

    [7]

    M. Jotz, Infinitesimal objects associated to Dirac groupoids and their homogeneous spaces, Preprint, arXiv:1009.0713. (2010).

    [8]

    _______, "Dirac Group(oid)s and Their Homogeneous Spaces," Ph. D. thesis, EPFL, Lausanne, 2011.

    [9]

    _______, Dirac Lie groups, Dirac homogeneous spaces and the Theorem of Drinfel'd, arXiv:0910.1538, to appear in "Indiana University Mathematics Journal'' (2011).

    [10]

    M. Jotz, T. S. Ratiu and J. Śniatycki, Singular Dirac reduction, Trans. Amer. Math. Soc., 363 (2011), 2967-3013.doi: 10.1090/S0002-9947-2011-05220-7.

    [11]

    M. Jotz, T. Ratiu and M. Zambon, Invariant frames for vector bundles and applications, Geometriae Dedicata, 158 (2011), 1-12.

    [12]

    K. C. H. Mackenzie, "Lie Groupoids and Lie Algebroids in Differential Geometry," London Mathematical Society Lecture Note Series, 124, Cambridge University Press, Cambridge, 1987.doi: 10.1017/CBO9780511661839.

    [13]

    _______, Double Lie algebroids and second-order geometry. II, Adv. Math., 154 (2000), 46-75.doi: 10.1006/aima.1999.1892.

    [14]

    _______, "General Theory of Lie Groupoids and Lie Algebroids," London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.

    [15]

    I. Moerdijk and J. Mrčun, "Introduction to Foliations and Lie Groupoids," Cambridge Studies in Advanced Mathematics. 91. Cambridge: Cambridge University Press. 2003. x+173 pp.

    [16]

    J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348. (English).doi: 10.1088/0951-7715/19/6/006.

    [17]

    J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics (Boston, Mass.), 222. Boston, MA: Birkhäuser. 2004, xxxiv+497 pp.

    [18]

    C. Ortiz, Multiplicative Dirac structures on Lie groups, C. R., Math., Acad. Sci. Paris, 346 (2008), 1279-1282.doi: 10.1016/j.crma.2008.10.003.

    [19]

    _______, "Multiplicative Dirac Structures," Ph. D. thesis, Instituto de Matemática Pura e Aplicada, 2009.

    [20]

    J. Pradines, Remarque sur le groupoïde cotangent de Weinstein-Dazord, C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 557-560.

    [21]

    P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. (3), 29 (1974), 699-713.doi: 10.1112/plms/s3-29.4.699.

    [22]

    _______, Integrability of systems of vector fields, J. London Math. Soc. (2), 21 (1980), 544-556.

    [23]

    H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc., 180 (1973), 171-188.doi: 10.1090/S0002-9947-1973-0321133-2.

    [24]

    A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705-727.doi: 10.2969/jmsj/04040705.

    [25]

    _______, Lagrangian mechanics and groupoids, Mechanics day (Waterloo, ON, 1992), 207-231, Fields Inst. Commun., 7, Amer. Math. Soc., Providence, RI, 1996.

    [26]

    M. Zambon, Reduction of branes in generalized complex geometry, J. Symplectic Geom., 6 (2008), 353-378.

    [27]

    _______, Submanifolds in poisson geometry: A survey, Complex and Differential Geometry, Springer Proceedings in Mathematics, Springer Berlin, 8 (2010), 403-420.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(95) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return