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Graded Bundles in the Category of Lie Groupoids
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numerous natural examples. Note that $\mathcal{VB}$-groupoids, extensively studied in the recent literature, form just the particular case of weighted Lie groupoids of degree one. We examine the Lie theory related to weighted groupoids and weighted Lie algebroids, objects defined in a previous publication of the authors, which are graded manifolds in the category of Lie algebroids, showing that they are naturally related via differentiation and integration. In this work we also make an initial study of weighted Poisson–Lie groupoids and weighted Lie bi-algebroids, as well as weighted Courant algebroids.
Keywords: graded manifolds; homogeneity structures; Lie groupoids; Lie algebroids. 
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Andrew James Bruce; Katarzyna Grabowska; Janusz Grabowski. Graded Bundles in the Category of Lie Groupoids. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a89/

[1] Brahic O., Cabrera A., Ortiz C., Obstructions to the integrability of $\mathcal{VB}$-algebroids, arXiv: 1403.1990

[2] Bruce A. J., Grabowska K., Grabowski J., “Higher order mechanics on graded bundles”, J. Phys. A: Math. Theor., 48 (2015), 205203, 32 pp., arXiv: 1412.2719 | DOI | MR | Zbl

[3] Bruce A. J., Grabowska K., Grabowski J., Linear duals of graded bundles and higher analogues of (Lie) algebroids, arXiv: 1409.0439

[4] Bruce A. J., Grabowska K., Grabowski J., Remarks on contact and Jacobi geometry, arXiv: 1507.05405

[5] Bursztyn H., Cabrera A., del Hoyo M., Vector bundles over Lie groupoids and algebroids, arXiv: 1410.5135

[6] Courant T. J., “Dirac manifolds”, Trans. Amer. Math. Soc., 319 (1990), 631–661 | DOI | MR | Zbl

[7] Crainic M., Fernandes R. L., “Integrability of Lie brackets”, Ann. of Math., 157 (2003), 575–620, arXiv: math.DG/0105033 | DOI | MR | Zbl

[8] Crainic M., Fernandes R. L., “Lectures on integrability of Lie brackets”, Lectures on Poisson geometry, Geom. Topol. Monogr., 17, Geom. Topol. Publ., Coventry, 2011, 1–107, arXiv: math.DG/0611259 | MR | Zbl

[9] Crainic M., Zhu C., “Integrability of Jacobi and Poisson structures”, Ann. Inst. Fourier (Grenoble), 57 (2007), 1181–1216, arXiv: math.DG/0403268 | DOI | MR | Zbl

[10] Dazord P., “Intégration d'algèbres de Lie locales et groupoïdes de contact”, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 959–964 | MR | Zbl

[11] Grabowski J., “Quasi-derivations and QD-algebroids”, Rep. Math. Phys., 52 (2003), 445–451, arXiv: math.DG/0301234 | DOI | MR | Zbl

[12] Grabowski J., “Graded contact manifolds and contact Courant algebroids”, J. Geom. Phys., 68 (2013), 27–58, arXiv: 1112.0759 | DOI | MR | Zbl

[13] Grabowski J., Grabowska K., Urbański P., “Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings”, J. Geom. Mech., 6 (2014), 503–526, arXiv: 1401.6970 | DOI | MR | Zbl

[14] Grabowski J., Rotkiewicz M., “Higher vector bundles and multi-graded symplectic manifolds”, J. Geom. Phys., 59 (2009), 1285–1305, arXiv: math.DG/0702772 | DOI | MR | Zbl

[15] Grabowski J., Rotkiewicz M., “Graded bundles and homogeneity structures”, J. Geom. Phys., 62 (2012), 21–36, arXiv: 1102.0180 | DOI | MR | Zbl

[16] Gracia-Saz A., Mehta R. A., “Lie algebroid structures on double vector bundles and representation theory of Lie algebroids”, Adv. Math., 223 (2010), 1236–1275, arXiv: 0810.0066 | DOI | MR | Zbl

[17] Gracia-Saz A., Mehta R. A., $\mathcal{VB}$-groupoids and representations of Lie groupoids, arXiv: 1007.3658

[18] Kerbrat Y., Souici-Benhammadi Z., “Variétés de {J}acobi et groupoïdes de contact”, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 81–86 | MR | Zbl

[19] Kirillov A. A., “Local Lie algebras”, Russ. Math. Surv., 31:4 (1976), 55–76 | DOI | MR

[20] Kolář I., Michor P. W., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993 | DOI | MR | Zbl

[21] Kontsevich M., “Deformation quantization of Poisson manifolds”, Lett. Math. Phys., 66 (2003), 157–216, arXiv: q-alg/9709040 | DOI | MR | Zbl

[22] Kosmann-Schwarzbach Y., “Exact Gerstenhaber algebras and Lie bialgebroids”, Acta Appl. Math., 41 (1995), 153–165 | DOI | MR | Zbl

[23] Kosmann-Schwarzbach Y., “Derived brackets”, Lett. Math. Phys., 69 (2004), 61–87, arXiv: math.DG/0312524 | DOI | MR | Zbl

[24] Kouotchop Wamba P. M., Ntyam A., “Tangent lifts of higher order of multiplicative Dirac structures”, Arch. Math. (Brno), 49 (2013), 87–104 | DOI | MR | Zbl

[25] Liu Z.-J., Weinstein A., Xu P., “Manin triples for Lie bialgebroids”, J. Differential Geom., 45 (1997), 547–574, arXiv: dg-ga/9508013 | MR | Zbl

[26] Mackenzie K. C. H., Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, 124, Cambridge University Press, Cambridge, 1987 | DOI | MR | Zbl

[27] Mackenzie K. C. H., “On symplectic double groupoids and the duality of Poisson groupoids”, Internat. J. Math., 10 (1999), 435–456, arXiv: math.DG/9808005 | DOI | MR | Zbl

[28] Mackenzie K. C. H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005 | DOI | MR | Zbl

[29] Mackenzie K. C. H., Xu P., “Lie bialgebroids and Poisson groupoids”, Duke Math. J., 73 (1994), 415–452 | DOI | MR | Zbl

[30] Mackenzie K. C. H., Xu P., “Integration of Lie bialgebroids”, Topology, 39 (2000), 445–467, arXiv: dg-ga/9712012 | DOI | MR | Zbl

[31] Marle C.-M., “On Jacobi manifolds and Jacobi bundles”, Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., 20, Springer, New York, 1991, 227–246 | DOI | MR

[32] Martínez E., “Higher-order variational calculus on Lie algebroids”, J. Geom. Mech., 7 (2015), 81–108, arXiv: 1501.0652 | DOI | MR | Zbl

[33] Mehta R. A., Supergroupoids, double structures, and equivariant cohomology, Ph.D. Thesis, University of California, Berkeley, 2006, arXiv: math.DG/0605356 | MR

[34] Mehta R. A., “$Q$-groupoids and their cohomology”, Pacific J. Math., 242 (2009), 311–332, arXiv: math.DG/0611924 | DOI | MR | Zbl

[35] Mehta R. A., “Differential graded contact geometry and Jacobi structures”, Lett. Math. Phys., 103 (2013), 729–741, arXiv: 1111.4705 | DOI | MR | Zbl

[36] Moerdijk I., Mrčun J., “On integrability of infinitesimal actions”, Amer. J. Math., 124 (2002), 567–593, arXiv: math.DG/0006042 | DOI | MR | Zbl

[37] Pradines J., “Représentation des jets non holonomes par des morphismes vectoriels doubles soudés”, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 1523–1526 | MR | Zbl

[38] Roytenberg D., “On the structure of graded symplectic supermanifolds and Courant algebroids”, Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., 315, Amer. Math. Soc., Providence, RI, 2002, 169–185, arXiv: math.SG/0203110 | DOI | MR | Zbl

[39] Roytenberg D., “AKSZ-BV formalism and Courant algebroid-induced topological field theories”, Lett. Math. Phys., 79 (2007), 143–159, arXiv: hep-th/0608150 | DOI | MR | Zbl

[40] Schwarz A., “Semiclassical approximation in Batalin–Vilkovisky formalism”, Comm. Math. Phys., 158 (1993), 373–396, arXiv: hep-th/9210115 | DOI | MR

[41] Ševera P., “Some title containing the words “homotopy” and “symplectic”, e.g. this one”, Trav. Math., 16 (2005), 121–137, arXiv: math.SG/0105080 | MR

[42] Tseng H.-H., Zhu C., “Integrating Lie algebroids via stacks”, Compos. Math., 142 (2006), 251–270, arXiv: math.DF/0405003 | DOI | MR | Zbl

[43] Vaintrob A. Yu., “Lie algebroids and homological vector fields”, Russ. Math. Surv., 52 (1997), 428–429 | DOI | MR | Zbl

[44] Voronov Th. Th., “Graded manifolds and Drinfeld doubles for Lie bialgebroids”, Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., 315, Amer. Math. Soc., Providence, RI, 2002, 131–168, arXiv: math.DG/0105237 | DOI | MR | Zbl

[45] Voronov Th. Th., “$Q$-manifolds and higher analogs of Lie algebroids”, XXIX Workshop on Geometric Methods in Physics, AIP Conf. Proc., 1307, Amer. Inst. Phys., Melville, NY, 2010, 191–202, arXiv: 1010.2503 | DOI | MR | Zbl

[46] Voronov Th. Th., “$Q$-manifolds and Mackenzie theory”, Comm. Math. Phys., 315 (2012), 279–310, arXiv: 1206.3622 | DOI | MR | Zbl