# 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. 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

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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. 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
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