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, 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.

show all references

References:
[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.

[1]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064

[2]

Víctor Manuel Jiménez Morales, Manuel De León, Marcelo Epstein. Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies. Journal of Geometric Mechanics, 2019, 11 (3) : 301-324. doi: 10.3934/jgm.2019017

[3]

Theodore Voronov. Book review: General theory of Lie groupoids and Lie algebroids, by Kirill C. H. Mackenzie. Journal of Geometric Mechanics, 2021, 13 (3) : 277-283. doi: 10.3934/jgm.2021026

[4]

Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009

[5]

Juan Carlos Marrero, David Martín de Diego, Ari Stern. Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367

[6]

Robert Lauter and Victor Nistor. On spectra of geometric operators on open manifolds and differentiable groupoids. Electronic Research Announcements, 2001, 7: 45-53.

[7]

Víctor Manuel Jiménez, Manuel de León. The evolution equation: An application of groupoids to material evolution. Journal of Geometric Mechanics, 2022, 14 (2) : 331-348. doi: 10.3934/jgm.2022001

[8]

Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo, Luca Schiavone. Schwinger's picture of quantum mechanics: 2-groupoids and symmetries. Journal of Geometric Mechanics, 2021, 13 (3) : 333-354. doi: 10.3934/jgm.2021008

[9]

Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007

[10]

Henrique Bursztyn, Alejandro Cabrera, Matias del Hoyo. Poisson double structures. Journal of Geometric Mechanics, 2022, 14 (2) : 151-178. doi: 10.3934/jgm.2021029

[11]

Misael Avendaño-Camacho, Isaac Hasse-Armengol, Eduardo Velasco-Barreras, Yury Vorobiev. The method of averaging for Poisson connections on foliations and its applications. Journal of Geometric Mechanics, 2020, 12 (3) : 343-361. doi: 10.3934/jgm.2020015

[12]

Dmitry Tamarkin. Quantization of Poisson structures on R^2. Electronic Research Announcements, 1997, 3: 119-120.

[13]

Mohammad Shafiee. The 2-plectic structures induced by the Lie bialgebras. Journal of Geometric Mechanics, 2017, 9 (1) : 83-90. doi: 10.3934/jgm.2017003

[14]

Nicola Sansonetto, Daniele Sepe. Twisted isotropic realisations of twisted Poisson structures. Journal of Geometric Mechanics, 2013, 5 (2) : 233-256. doi: 10.3934/jgm.2013.5.233

[15]

Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247

[16]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421

[17]

Dennis I. Barrett, Rory Biggs, Claudiu C. Remsing, Olga Rossi. Invariant nonholonomic Riemannian structures on three-dimensional Lie groups. Journal of Geometric Mechanics, 2016, 8 (2) : 139-167. doi: 10.3934/jgm.2016001

[18]

Andrey Tsiganov. Poisson structures for two nonholonomic systems with partially reduced symmetries. Journal of Geometric Mechanics, 2014, 6 (3) : 417-440. doi: 10.3934/jgm.2014.6.417

[19]

Fang Li, Jie Pan. On inner Poisson structures of a quantum cluster algebra without coefficients. Electronic Research Archive, 2021, 29 (5) : 2959-2972. doi: 10.3934/era.2021021

[20]

Pengliang Xu, Xiaomin Tang. Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra. Electronic Research Archive, 2021, 29 (4) : 2771-2789. doi: 10.3934/era.2021013

2021 Impact Factor: 0.737

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]