September  2012, 4(3): 313-332. doi: 10.3934/jgm.2012.4.313

The leaf space of a multiplicative foliation

1. 

Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

Received  November 2010 Revised  October 2011 Published  October 2012

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.
Citation: M. Jotz. The leaf space of a multiplicative foliation. Journal of Geometric Mechanics, 2012, 4 (3) : 313-332. doi: 10.3934/jgm.2012.4.313
References:
[1]

A. Coste, P. Dazord and A. Weinstein, Groupoï des symplectiques,, Publications du Département de Mathématiques. Nouvelle Série. A, 2 (1987), 1. Google Scholar

[2]

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

[3]

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

[4]

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

[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. doi: 10.1007/s00332-007-9012-8. Google Scholar

[6]

M. Jotz and C. Ortiz, Foliated groupoids and their infinitesimal data,, Preprint, (2011). Google Scholar

[7]

M. Jotz, Infinitesimal objects associated to Dirac groupoids and their homogeneous spaces,, Preprint, (2010). Google Scholar

[8]

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

[9]

_______, Dirac Lie groups, Dirac homogeneous spaces and the Theorem of Drinfel'd,, , (2011). Google Scholar

[10]

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

[11]

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

[12]

K. C. H. Mackenzie, "Lie Groupoids and Lie Algebroids in Differential Geometry,", London Mathematical Society Lecture Note Series, 124 (1987). doi: 10.1017/CBO9780511661839. Google Scholar

[13]

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

[14]

_______, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series, 213 (2005). Google Scholar

[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., 91 (2003). Google Scholar

[16]

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

[17]

J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics (Boston, 222 (2004). Google Scholar

[18]

C. Ortiz, Multiplicative Dirac structures on Lie groups,, C. R., 346 (2008), 1279. doi: 10.1016/j.crma.2008.10.003. Google Scholar

[19]

_______, "Multiplicative Dirac Structures,", Ph. D. thesis, (2009). Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

_______, Lagrangian mechanics and groupoids,, Mechanics day (Waterloo, (1992), 207. Google Scholar

[26]

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

[27]

_______, Submanifolds in poisson geometry: A survey,, Complex and Differential Geometry, 8 (2010), 403. Google Scholar

show all references

References:
[1]

A. Coste, P. Dazord and A. Weinstein, Groupoï des symplectiques,, Publications du Département de Mathématiques. Nouvelle Série. A, 2 (1987), 1. Google Scholar

[2]

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

[3]

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

[4]

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

[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. doi: 10.1007/s00332-007-9012-8. Google Scholar

[6]

M. Jotz and C. Ortiz, Foliated groupoids and their infinitesimal data,, Preprint, (2011). Google Scholar

[7]

M. Jotz, Infinitesimal objects associated to Dirac groupoids and their homogeneous spaces,, Preprint, (2010). Google Scholar

[8]

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

[9]

_______, Dirac Lie groups, Dirac homogeneous spaces and the Theorem of Drinfel'd,, , (2011). Google Scholar

[10]

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

[11]

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

[12]

K. C. H. Mackenzie, "Lie Groupoids and Lie Algebroids in Differential Geometry,", London Mathematical Society Lecture Note Series, 124 (1987). doi: 10.1017/CBO9780511661839. Google Scholar

[13]

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

[14]

_______, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series, 213 (2005). Google Scholar

[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., 91 (2003). Google Scholar

[16]

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

[17]

J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics (Boston, 222 (2004). Google Scholar

[18]

C. Ortiz, Multiplicative Dirac structures on Lie groups,, C. R., 346 (2008), 1279. doi: 10.1016/j.crma.2008.10.003. Google Scholar

[19]

_______, "Multiplicative Dirac Structures,", Ph. D. thesis, (2009). Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

_______, Lagrangian mechanics and groupoids,, Mechanics day (Waterloo, (1992), 207. Google Scholar

[26]

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

[27]

_______, Submanifolds in poisson geometry: A survey,, Complex and Differential Geometry, 8 (2010), 403. Google Scholar

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