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Lie Algebroids in the Loday–Pirashvili Category
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 describe Lie–Rinehart algebras in the tensor category $\mathcal{LM}$ of linear maps in the sense of Loday and Pirashvili and construct a functor from Lie–Rinehart algebras in $\mathcal{LM}$ to Leibniz algebroids.
Mots-clés : Lie algebroid; Leibniz algebra; Courant algebroid; Leibniz algebroid. 
@article{SIGMA_2015_11_a78,
     author = {Ana Rovi},
     title = {Lie {Algebroids} in the {Loday{\textendash}Pirashvili} {Category}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a78/}
}
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Ana Rovi. Lie Algebroids in the Loday–Pirashvili Category. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a78/

[1] Baraglia D., “Leibniz algebroids, twistings and exceptional generalized geometry”, J. Geom. Phys., 62 (2012), 903–934, arXiv: 1101.0856 | DOI | MR | Zbl

[2] Bloh A., “On a generalization of the concept of Lie algebra”, Dokl. Akad. Nauk SSSR, 165 (1965), 471–473 | MR

[3] Böhm G., Szlachányi K., “Hopf algebroids with bijective antipodes: axioms, integrals, and duals”, J. Algebra, 274 (2004), 708–750, arXiv: math.QA/0302325 | DOI | MR | Zbl

[4] Cuvier C., “Homologie de Leibniz et homologie de Hochschild”, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), 569–572 | MR | Zbl

[5] Figueroa-O'Farrill J. M., “Deformations of 3-algebras”, J. Math. Phys., 50 (2009), 113514, 27 pp., arXiv: 0903.4871 | DOI | MR | Zbl

[6] Herz J.-C., “Pseudo-algèbres de Lie, I”, C. R. Acad. Sci. Paris, 236 (1953), 1935–1937 | MR | Zbl

[7] Huebschmann J., “Poisson cohomology and quantization”, J. Reine Angew. Math., 408 (1990), 57–113 | DOI | MR | Zbl

[8] Huebschmann J., “On the quantization of Poisson algebras”, Symplectic Geometry and Mathematical Physics (Aix-en-Provence, 1990), Progr. Math., 99, Birkhäuser Boston, Boston, MA, 1991, 204–233 | MR

[9] Huebschmann J., “Lie–Rinehart algebras, Gerstenhaber algebras and Batalin–Vilkovisky algebras”, Ann. Inst. Fourier (Grenoble), 48 (1998), 425–440, arXiv: dg-ga/9704005 | DOI | MR | Zbl

[10] Ibáñez R., de León M., Marrero J. C., Padrón E., “Leibniz algebroid associated with a Nambu–Poisson structure”, J. Phys. A: Math. Gen., 32 (1999), 8129–8144, arXiv: math-ph/9906027 | DOI | MR | Zbl

[11] Kinyon M. K., Weinstein A., “Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces”, Amer. J. Math., 123 (2001), 525–550, arXiv: math.DG/0006022 | DOI | MR | Zbl

[12] Kosmann-Schwarzbach Y., “Courant algebroids. A short history”, SIGMA, 9 (2013), 014, 8 pp., arXiv: 1212.0559 | DOI | MR | Zbl

[13] Krähmer U., Rovi A., “A Lie–Rinehart algebra with no antipode”, Comm. Algebra, 43 (2015), 4049–4053 | DOI | MR | Zbl

[14] Krähmer U., Wagemann F., “Racks, Leibniz algebras and Yetter–Drinfel'd modules”, Georgian Math. J. (to appear)

[15] Kurdiani R., Pirashvili T., “A Leibniz algebra structure on the second tensor power”, J. Lie Theory, 12 (2002), 583–596 | MR | Zbl

[16] 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

[17] Loday J.-L., “Une version non commutative des algèbres de Lie: les algèbres de Leibniz”, Enseign. Math., 39 (1993), 269–293 | MR | Zbl

[18] Loday J.-L., Pirashvili T., “Universal enveloping algebras of Leibniz algebras and (co)homology”, Math. Ann., 296 (1993), 139–158 | DOI | MR | Zbl

[19] Loday J.-L., Pirashvili T., “The tensor category of linear maps and Leibniz algebras”, Georgian Math. J., 5 (1998), 263–276 | DOI | MR | Zbl

[20] Pradines J., “Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie des groupoïdes infinitésimaux”, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A245–A248 | MR

[21] Rinehart G. S., “Differential forms on general commutative algebras”, Trans. Amer. Math. Soc., 108 (1963), 195–222 | DOI | MR | Zbl

[22] Rovi A., “Hopf algebroids associated to Jacobi algebras”, Int. J. Geom. Methods Mod. Phys., 11 (2014), 1450092, 20 pp., arXiv: 1311.4181 | DOI | MR

[23] Roytenberg D., “Courant–Dorfman algebras and their cohomology”, Lett. Math. Phys., 90 (2009), 311–351, arXiv: 0902.4862 | DOI | MR | Zbl

[24] Stiénon M., Xu P., “Modular classes of Loday algebroids”, C. R. Math. Acad. Sci. Paris, 346 (2008), 193–198, arXiv: 0803.2047 | DOI | MR | Zbl