December  2012, 4(4): 469-485. doi: 10.3934/jgm.2012.4.469

Distributions and quotients on degree $1$ NQ-manifolds and Lie algebroids

1. 

Universidad Autónoma de Madrid (Dept. de Matemáticas), ICMAT(CSIC-UAM-UC3M-UCM), Campus de Cantoblanco, 28049 - Madrid, Spain

2. 

Courant Research Centre “Higher Order Structures”, Mathematisches Institut, University of Göttingen, Göttingen, 37073, Germany

Received  July 2012 Revised  August 2012 Published  January 2013

It is well-known that a Lie algebroid $A$ is equivalently described by a degree 1 NQ-manifold $\mathcal{M}$. We study distributions on $\mathcal{M}$, giving a characterization in terms of $A$. We show that involutive $Q$-invariant distributions on $\mathcal{M}$ correspond bijectively to IM-foliations on $A$ (the infinitesimal version of Mackenzie's ideal systems). We perform reduction by such distributions, and investigate how they arise from non-strict actions of strict Lie 2-algebras on $\mathcal{M}$.
Citation: Marco Zambon, Chenchang Zhu. Distributions and quotients on degree $1$ NQ-manifolds and Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 469-485. doi: 10.3934/jgm.2012.4.469
References:
[1]

J. C. Baez and A. S. Crans, Higher-dimensional algebra. VI. Lie 2-algebras,, Theory Appl. Categ., 12 (2004), 492. Google Scholar

[2]

O. Brahic and C. Zhu, Lie algebroid fibrations,, Adv. Math., (2010). doi: 10.1016/j.aim.2010.10.006. Google Scholar

[3]

H. Bursztyn, A. S. Cattaneo, R. Metha and M. Zambon, Reduction of Courant algebroids via super-geometry,, in preparation., (). Google Scholar

[4]

H. Bursztyn and M. Crainic, Dirac structures, momentum maps, and quasi-Poisson manifolds,, in, 232 (2005), 1. doi: 10.1007/0-8176-4419-9_1. Google Scholar

[5]

A. S. Cattaneo, From topological field theory to deformation quantization and reduction,, in, III (2006), 339. Google Scholar

[6]

A. S. Cattaneo and F. Schätz, Introduction to supergeometry,, Rev. Math. Phys., 23 (2011), 669. doi: 10.1142/S0129055X11004400. Google Scholar

[7]

A. S. Cattaneo and M. Zambon, A super-geometric approach to Poisson reduction,, To appear in Comm. Math. Physics., (). Google Scholar

[8]

M. Jotz and C. Ortiz, Foliated groupoids and their infinitesimal data,, \arXiv{1109.4515}., (). Google Scholar

[9]

Y. Kosmann-Schwarzbach, Derived brackets,, Lett. Math. Phys., 69 (2004), 61. doi: 10.1007/s11005-004-0608-8. Google Scholar

[10]

Y. Kosmann-Schwarzbach and K. C. H. Mackenzie, Differential operators and actions of Lie algebroids,, in, 315 (2002), 213. doi: 10.1090/conm/315/05482. Google Scholar

[11]

T. Lada and M. Markl, Strongly homotopy Lie algebras,, Comm. Algebra, 23 (1995), 2147. doi: 10.1080/00927879508825335. Google Scholar

[12]

K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,'', \textbf{213} of London Mathematical Society Lecture Note Series. Cambridge University Press, 213 (2005). Google Scholar

[13]

R. A. Mehta, "Supergroupoids, Double Structures, and Equivariant Cohomology,'', Ph.D thesis, (2006). Google Scholar

[14]

R. A. Mehta and M. Zambon, $L_{\infty}$-algebra actions on graded manifolds,, to appear in Differential Geometry and its Applications., (). doi: 10.1016/j.difgeo.2012.07.006. Google Scholar

[15]

P. Ševera, Letter to Alan Weinstein,, \url{http://sophia.dtp.fmph.uniba.sk/~severa/letters/no8.ps}., (). Google Scholar

[16]

P. Ševera, Some title containing the words "homotopy'' and "symplectic'', e.g. this one,, in, (2005), 121. Google Scholar

[17]

P. Ševera, Poisson actions up to homotopy and their quantization,, Lett. Math. Phys., 77 (2006), 199. doi: 10.1007/s11005-006-0089-z. Google Scholar

[18]

L. Stefanini, "On Morphic Actions and Integrability of LA-Groupoids,'', Ph.D thesis, (2009). Google Scholar

[19]

A. Y. Vaĭntrob, Lie algebroids and homological vector fields,, Uspekhi Mat. Nauk, 52 (1997), 161. doi: 10.1070/RM1997v052n02ABEH001802. Google Scholar

[20]

T. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids,, in, 315 (2002), 131. doi: 10.1090/conm/315/05478. Google Scholar

[21]

T. Voronov, Mackenzie theory and Q-manifolds,, \arXiv{math/0608111}, (2006). Google Scholar

[22]

T. T. Voronov, Q-manifolds and higher analogs of Lie algebroids,, XXIX Workshop on Geometric Methods in Physics. AIP CP 1307, (2010), 191. Google Scholar

[23]

M. Zambon and C. Zhu, Higher Lie algebra actions on Lie algebroids,, \arXiv{1012.0428v2} to appear in Journal of Geometry and Physics., (). Google Scholar

show all references

References:
[1]

J. C. Baez and A. S. Crans, Higher-dimensional algebra. VI. Lie 2-algebras,, Theory Appl. Categ., 12 (2004), 492. Google Scholar

[2]

O. Brahic and C. Zhu, Lie algebroid fibrations,, Adv. Math., (2010). doi: 10.1016/j.aim.2010.10.006. Google Scholar

[3]

H. Bursztyn, A. S. Cattaneo, R. Metha and M. Zambon, Reduction of Courant algebroids via super-geometry,, in preparation., (). Google Scholar

[4]

H. Bursztyn and M. Crainic, Dirac structures, momentum maps, and quasi-Poisson manifolds,, in, 232 (2005), 1. doi: 10.1007/0-8176-4419-9_1. Google Scholar

[5]

A. S. Cattaneo, From topological field theory to deformation quantization and reduction,, in, III (2006), 339. Google Scholar

[6]

A. S. Cattaneo and F. Schätz, Introduction to supergeometry,, Rev. Math. Phys., 23 (2011), 669. doi: 10.1142/S0129055X11004400. Google Scholar

[7]

A. S. Cattaneo and M. Zambon, A super-geometric approach to Poisson reduction,, To appear in Comm. Math. Physics., (). Google Scholar

[8]

M. Jotz and C. Ortiz, Foliated groupoids and their infinitesimal data,, \arXiv{1109.4515}., (). Google Scholar

[9]

Y. Kosmann-Schwarzbach, Derived brackets,, Lett. Math. Phys., 69 (2004), 61. doi: 10.1007/s11005-004-0608-8. Google Scholar

[10]

Y. Kosmann-Schwarzbach and K. C. H. Mackenzie, Differential operators and actions of Lie algebroids,, in, 315 (2002), 213. doi: 10.1090/conm/315/05482. Google Scholar

[11]

T. Lada and M. Markl, Strongly homotopy Lie algebras,, Comm. Algebra, 23 (1995), 2147. doi: 10.1080/00927879508825335. Google Scholar

[12]

K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,'', \textbf{213} of London Mathematical Society Lecture Note Series. Cambridge University Press, 213 (2005). Google Scholar

[13]

R. A. Mehta, "Supergroupoids, Double Structures, and Equivariant Cohomology,'', Ph.D thesis, (2006). Google Scholar

[14]

R. A. Mehta and M. Zambon, $L_{\infty}$-algebra actions on graded manifolds,, to appear in Differential Geometry and its Applications., (). doi: 10.1016/j.difgeo.2012.07.006. Google Scholar

[15]

P. Ševera, Letter to Alan Weinstein,, \url{http://sophia.dtp.fmph.uniba.sk/~severa/letters/no8.ps}., (). Google Scholar

[16]

P. Ševera, Some title containing the words "homotopy'' and "symplectic'', e.g. this one,, in, (2005), 121. Google Scholar

[17]

P. Ševera, Poisson actions up to homotopy and their quantization,, Lett. Math. Phys., 77 (2006), 199. doi: 10.1007/s11005-006-0089-z. Google Scholar

[18]

L. Stefanini, "On Morphic Actions and Integrability of LA-Groupoids,'', Ph.D thesis, (2009). Google Scholar

[19]

A. Y. Vaĭntrob, Lie algebroids and homological vector fields,, Uspekhi Mat. Nauk, 52 (1997), 161. doi: 10.1070/RM1997v052n02ABEH001802. Google Scholar

[20]

T. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids,, in, 315 (2002), 131. doi: 10.1090/conm/315/05478. Google Scholar

[21]

T. Voronov, Mackenzie theory and Q-manifolds,, \arXiv{math/0608111}, (2006). Google Scholar

[22]

T. T. Voronov, Q-manifolds and higher analogs of Lie algebroids,, XXIX Workshop on Geometric Methods in Physics. AIP CP 1307, (2010), 191. Google Scholar

[23]

M. Zambon and C. Zhu, Higher Lie algebra actions on Lie algebroids,, \arXiv{1012.0428v2} to appear in Journal of Geometry and Physics., (). Google Scholar

[1]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81

[2]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-15. doi: 10.3934/dcdss.2020066

[3]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[4]

Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213

[5]

Dennise García-Beltrán, José A. Vallejo, Yurii Vorobiev. Lie algebroids generated by cohomology operators. Journal of Geometric Mechanics, 2015, 7 (3) : 295-315. doi: 10.3934/jgm.2015.7.295

[6]

K. C. H. Mackenzie. Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids. Electronic Research Announcements, 1998, 4: 74-87.

[7]

Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004

[8]

Felipe A. Ramírez. Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups. Journal of Modern Dynamics, 2009, 3 (3) : 335-357. doi: 10.3934/jmd.2009.3.335

[9]

Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239

[10]

Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453

[11]

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

[12]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213

[13]

Juan Carlos Marrero. Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes. Journal of Geometric Mechanics, 2010, 2 (3) : 243-263. doi: 10.3934/jgm.2010.2.243

[14]

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

[15]

Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451

[16]

Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39

[17]

André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351

[18]

Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517

[19]

Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, 2017, 9 (1) : 1-45. doi: 10.3934/jgm.2017001

[20]

Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105

2018 Impact Factor: 0.525

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]