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

Previous Article
JGM Home
This Issue
Next Article
The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity

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

Abstract
Related Papers

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]	J. C. Baez and A. S. Crans, Higher-dimensional algebra. VI. Lie 2-algebras, Theory Appl. Categ., 12 (2004), 492-538 (electronic).
[2]	O. Brahic and C. Zhu, Lie algebroid fibrations, Adv. Math., 2010, arXiv:1001.4904. doi: 10.1016/j.aim.2010.10.006.
[3]	H. Bursztyn, A. S. Cattaneo, R. Metha and M. Zambon, Reduction of Courant algebroids via super-geometry, in preparation.
[4]	H. Bursztyn and M. Crainic, Dirac structures, momentum maps, and quasi-Poisson manifolds, 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.
[5]	A. S. Cattaneo, From topological field theory to deformation quantization and reduction, in "International Congress of Mathematicians'' III, 339-365. Eur. Math. Soc., Zürich, (2006).
[6]	A. S. Cattaneo and F. Schätz, Introduction to supergeometry, Rev. Math. Phys., 23 (2011), 669-690. doi: 10.1142/S0129055X11004400.
[7]	A. S. Cattaneo and M. Zambon, A super-geometric approach to Poisson reduction, To appear in Comm. Math. Physics.
[8]	M. Jotz and C. Ortiz, Foliated groupoids and their infinitesimal data, arXiv:1109.4515.
[9]	Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87. doi: 10.1007/s11005-004-0608-8.
[10]	Y. Kosmann-Schwarzbach and K. C. H. Mackenzie, Differential operators and actions of Lie algebroids, 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.
[11]	T. Lada and M. Markl, Strongly homotopy Lie algebras, Comm. Algebra, 23 (1995), 2147-2161. doi: 10.1080/00927879508825335.
[12]	K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,'' 213 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2005.
[13]	R. A. Mehta, "Supergroupoids, Double Structures, and Equivariant Cohomology,'' Ph.D thesis, University of California, Berkeley, 2006. arXiv:math.DG/0605356.
[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.
[15]	P. Ševera, Letter to Alan Weinstein, http://sophia.dtp.fmph.uniba.sk/~severa/letters/no8.ps.
[16]	P. Ševera, Some title containing the words "homotopy'' and "symplectic'', e.g. this one, in "Travaux Mathématiques. Fasc. XVI'' Trav. Math., XVI, 121-137. Univ. Luxemb., Luxembourg, (2005).
[17]	P. Ševera, Poisson actions up to homotopy and their quantization, Lett. Math. Phys., 77 (2006), 199-208. doi: 10.1007/s11005-006-0089-z.
[18]	L. Stefanini, "On Morphic Actions and Integrability of LA-Groupoids,'' Ph.D thesis, arXiv:0902.2228, 2009.
[19]	A. Y. Vaĭntrob, Lie algebroids and homological vector fields, Uspekhi Mat. Nauk, 52 (1997), 161-162. doi: 10.1070/RM1997v052n02ABEH001802.
[20]	T. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, 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.
[21]	T. Voronov, Mackenzie theory and Q-manifolds, arXiv:math/0608111, (2006).
[22]	T. T. Voronov, Q-manifolds and higher analogs of Lie algebroids, XXIX Workshop on Geometric Methods in Physics. AIP CP 1307, 191-202, Amer. Inst. Phys., Melville, NY, (2010).
[23]	M. Zambon and C. Zhu, Higher Lie algebra actions on Lie algebroids, arXiv:1012.0428v2 to appear in Journal of Geometry and Physics.

show all references

References:

[1]	J. C. Baez and A. S. Crans, Higher-dimensional algebra. VI. Lie 2-algebras, Theory Appl. Categ., 12 (2004), 492-538 (electronic).
[2]	O. Brahic and C. Zhu, Lie algebroid fibrations, Adv. Math., 2010, arXiv:1001.4904. doi: 10.1016/j.aim.2010.10.006.
[3]	H. Bursztyn, A. S. Cattaneo, R. Metha and M. Zambon, Reduction of Courant algebroids via super-geometry, in preparation.
[4]	H. Bursztyn and M. Crainic, Dirac structures, momentum maps, and quasi-Poisson manifolds, 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.
[5]	A. S. Cattaneo, From topological field theory to deformation quantization and reduction, in "International Congress of Mathematicians'' III, 339-365. Eur. Math. Soc., Zürich, (2006).
[6]	A. S. Cattaneo and F. Schätz, Introduction to supergeometry, Rev. Math. Phys., 23 (2011), 669-690. doi: 10.1142/S0129055X11004400.
[7]	A. S. Cattaneo and M. Zambon, A super-geometric approach to Poisson reduction, To appear in Comm. Math. Physics.
[8]	M. Jotz and C. Ortiz, Foliated groupoids and their infinitesimal data, arXiv:1109.4515.
[9]	Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87. doi: 10.1007/s11005-004-0608-8.
[10]	Y. Kosmann-Schwarzbach and K. C. H. Mackenzie, Differential operators and actions of Lie algebroids, 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.
[11]	T. Lada and M. Markl, Strongly homotopy Lie algebras, Comm. Algebra, 23 (1995), 2147-2161. doi: 10.1080/00927879508825335.
[12]	K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,'' 213 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2005.
[13]	R. A. Mehta, "Supergroupoids, Double Structures, and Equivariant Cohomology,'' Ph.D thesis, University of California, Berkeley, 2006. arXiv:math.DG/0605356.
[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.
[15]	P. Ševera, Letter to Alan Weinstein, http://sophia.dtp.fmph.uniba.sk/~severa/letters/no8.ps.
[16]	P. Ševera, Some title containing the words "homotopy'' and "symplectic'', e.g. this one, in "Travaux Mathématiques. Fasc. XVI'' Trav. Math., XVI, 121-137. Univ. Luxemb., Luxembourg, (2005).
[17]	P. Ševera, Poisson actions up to homotopy and their quantization, Lett. Math. Phys., 77 (2006), 199-208. doi: 10.1007/s11005-006-0089-z.
[18]	L. Stefanini, "On Morphic Actions and Integrability of LA-Groupoids,'' Ph.D thesis, arXiv:0902.2228, 2009.
[19]	A. Y. Vaĭntrob, Lie algebroids and homological vector fields, Uspekhi Mat. Nauk, 52 (1997), 161-162. doi: 10.1070/RM1997v052n02ABEH001802.
[20]	T. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, 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.
[21]	T. Voronov, Mackenzie theory and Q-manifolds, arXiv:math/0608111, (2006).
[22]	T. T. Voronov, Q-manifolds and higher analogs of Lie algebroids, XXIX Workshop on Geometric Methods in Physics. AIP CP 1307, 191-202, Amer. Inst. Phys., Melville, NY, (2010).
[23]	M. Zambon and C. Zhu, Higher Lie algebra actions on Lie algebroids, arXiv:1012.0428v2 to appear in Journal of Geometry and Physics.

[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 and 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 and 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]	Juan Carlos Marrero, David Martín de Diego, Eduardo Martínez. Local convexity for second order differential equations on a Lie algebroid. Journal of Geometric Mechanics, 2021, 13 (3) : 477-499. doi: 10.3934/jgm.2021021
[9]	Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004
[10]	Madeleine Jotz Lean, Kirill C. H. Mackenzie. Transitive double Lie algebroids via core diagrams. Journal of Geometric Mechanics, 2021, 13 (3) : 403-457. doi: 10.3934/jgm.2021023
[11]	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
[12]	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
[13]	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
[14]	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
[15]	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
[16]	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
[17]	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
[18]	Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453
[19]	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
[20]	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

2021 Impact Factor: 0.737

Tools

Metrics

Other articles
by authors

on AIMS
Marco Zambon Chenchang Zhu
on Google Scholar
Marco Zambon Chenchang Zhu

[Back to Top]