American Institute of Mathematical Sciences

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

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

Marco Zambon 1, and  Chenchang Zhu 2,

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 $\mathcal{M}$.
Keywords: quotients of Lie algebroids., NQ-manifold, higher Lie algebra action, Lie algebroid, ideal system.
Mathematics Subject Classification: Primary: 53D17, 58A50, 18D3.
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]

Theory Appl. Categ., 12 (2004), 492-538 (electronic).  Google Scholar

[2]

Adv. Math., 2010, arXiv:1001.4904. 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]

in "The Breadth of Symplectic and Poisson Geometry,'' 232 of Progr. Math., 1-40. Birkhäuser Boston, Boston, MA, (2005). doi: 10.1007/0-8176-4419-9_1.  Google Scholar

[5]

in "International Congress of Mathematicians'' III, 339-365. Eur. Math. Soc., Zürich, (2006).  Google Scholar

[6]

Rev. Math. Phys., 23 (2011), 669-690. 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]

Lett. Math. Phys., 69 (2004), 61-87. doi: 10.1007/s11005-004-0608-8.  Google Scholar

[10]

in "Quantization, Poisson Brackets and Beyond'' (Manchester, 2001), 315 of Contemp. Math., 213-233. Amer. Math. Soc., Providence, RI, (2002). doi: 10.1090/conm/315/05482.  Google Scholar

[11]

Comm. Algebra, 23 (1995), 2147-2161. doi: 10.1080/00927879508825335.  Google Scholar

[12]

213 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2005.  Google Scholar

[13]

Ph.D thesis, University of California, Berkeley, 2006. arXiv:math.DG/0605356.  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]

in "Travaux Mathématiques. Fasc. XVI'' Trav. Math., XVI, 121-137. Univ. Luxemb., Luxembourg, (2005).  Google Scholar

[17]

Lett. Math. Phys., 77 (2006), 199-208. doi: 10.1007/s11005-006-0089-z.  Google Scholar

[18]

Ph.D thesis, arXiv:0902.2228, 2009. Google Scholar

[19]

Uspekhi Mat. Nauk, 52 (1997), 161-162. doi: 10.1070/RM1997v052n02ABEH001802.  Google Scholar

[20]

in "Quantization, Poisson brackets and beyond (Manchester, 2001),'' 315 of Contemp. Math., 131-168. Amer. Math. Soc., Providence, RI, (2002). doi: 10.1090/conm/315/05478.  Google Scholar

[21]

arXiv:math/0608111, (2006). Google Scholar

[22]

XXIX Workshop on Geometric Methods in Physics. AIP CP 1307, 191-202, Amer. Inst. Phys., Melville, NY, (2010).  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]

Theory Appl. Categ., 12 (2004), 492-538 (electronic).  Google Scholar

[2]

Adv. Math., 2010, arXiv:1001.4904. 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]

in "The Breadth of Symplectic and Poisson Geometry,'' 232 of Progr. Math., 1-40. Birkhäuser Boston, Boston, MA, (2005). doi: 10.1007/0-8176-4419-9_1.  Google Scholar

[5]

in "International Congress of Mathematicians'' III, 339-365. Eur. Math. Soc., Zürich, (2006).  Google Scholar

[6]

Rev. Math. Phys., 23 (2011), 669-690. 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]

Lett. Math. Phys., 69 (2004), 61-87. doi: 10.1007/s11005-004-0608-8.  Google Scholar

[10]

in "Quantization, Poisson Brackets and Beyond'' (Manchester, 2001), 315 of Contemp. Math., 213-233. Amer. Math. Soc., Providence, RI, (2002). doi: 10.1090/conm/315/05482.  Google Scholar

[11]

Comm. Algebra, 23 (1995), 2147-2161. doi: 10.1080/00927879508825335.  Google Scholar

[12]

213 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2005.  Google Scholar

[13]

Ph.D thesis, University of California, Berkeley, 2006. arXiv:math.DG/0605356.  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]

in "Travaux Mathématiques. Fasc. XVI'' Trav. Math., XVI, 121-137. Univ. Luxemb., Luxembourg, (2005).  Google Scholar

[17]

Lett. Math. Phys., 77 (2006), 199-208. doi: 10.1007/s11005-006-0089-z.  Google Scholar

[18]

Ph.D thesis, arXiv:0902.2228, 2009. Google Scholar

[19]

Uspekhi Mat. Nauk, 52 (1997), 161-162. doi: 10.1070/RM1997v052n02ABEH001802.  Google Scholar

[20]

in "Quantization, Poisson brackets and beyond (Manchester, 2001),'' 315 of Contemp. Math., 131-168. Amer. Math. Soc., Providence, RI, (2002). doi: 10.1090/conm/315/05478.  Google Scholar

[21]

arXiv:math/0608111, (2006). Google Scholar

[22]

XXIX Workshop on Geometric Methods in Physics. AIP CP 1307, 191-202, Amer. Inst. Phys., Melville, NY, (2010).  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]

Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017
[3]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1115-1129. doi: 10.3934/dcdss.2020066
[4]

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

[5]

Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213
[6]

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
[7]

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

[8]

Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004
[9]

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
[10]

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
[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]

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
[15]

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
[16]

Pengliang Xu, Xiaomin Tang. Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra. Electronic Research Archive, , () : -. doi: 10.3934/era.2021013
[17]

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
[18]

Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021020
[19]

Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39
[20]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences