# American Institute of Mathematical Sciences

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), 492-538 (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.

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##### 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.
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