[Cohomologie de déformation d'un algèbroïde de Lie et équivalence de Morita]
Soit un algébroïde de Lie. Dans cette note, nous prouvons qu'un pull-back de A lelong d'une fibration ayant des fibres homologiquement m-connexes possède la même cohomologie de déformation que A jusqu'au degré m.
Let be a Lie algebroid. In this short note, we prove that a pull-back of A along a fibration with homologically m-connected fibers shares the same deformation cohomology of A up to degree m.
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@article{CRMATH_2018__356_4_376_0,
author = {Sparano, Giovanni and Vitagliano, Luca},
title = {Deformation cohomology of {Lie} algebroids and {Morita} equivalence},
journal = {Comptes Rendus. Math\'ematique},
pages = {376--381},
publisher = {Elsevier},
volume = {356},
number = {4},
year = {2018},
doi = {10.1016/j.crma.2018.03.004},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2018.03.004/}
}
TY - JOUR AU - Sparano, Giovanni AU - Vitagliano, Luca TI - Deformation cohomology of Lie algebroids and Morita equivalence JO - Comptes Rendus. Mathématique PY - 2018 SP - 376 EP - 381 VL - 356 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2018.03.004/ DO - 10.1016/j.crma.2018.03.004 LA - en ID - CRMATH_2018__356_4_376_0 ER -
%0 Journal Article %A Sparano, Giovanni %A Vitagliano, Luca %T Deformation cohomology of Lie algebroids and Morita equivalence %J Comptes Rendus. Mathématique %D 2018 %P 376-381 %V 356 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2018.03.004/ %R 10.1016/j.crma.2018.03.004 %G en %F CRMATH_2018__356_4_376_0
Sparano, Giovanni; Vitagliano, Luca. Deformation cohomology of Lie algebroids and Morita equivalence. Comptes Rendus. Mathématique, Tome 356 (2018) no. 4, pp. 376-381. doi : 10.1016/j.crma.2018.03.004. http://www.numdam.org/articles/10.1016/j.crma.2018.03.004/
[1] Representations up to homotopy of Lie algebroids, J. Reine Angew. Math., Volume 663 (2012), pp. 91-126 (e-print) | arXiv
[2] Differentiable stacks and gerbes, J. Symplectic Geom., Volume 9 (2011), pp. 285-341 (e-print) | arXiv
[3] Van Est isomorphism for homogeneous cochains, Pac. J. Math., Volume 287 (2017), pp. 297-336 (e-print) | arXiv
[4] Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes, Comment. Math. Helv., Volume 78 (2003), pp. 681-721 (e-print) | arXiv
[5] Deformations of Lie groupoids, 2015 (e-print) | arXiv
[6] Foliation groupoids and their cyclic homology, Adv. Math., Volume 1157 (2001), pp. 177-197 (e-print) | arXiv
[7] Deformations of Lie brackets: cohomological aspects, J. Eur. Math. Soc., Volume 10 (2008), pp. 1037-1059 (e-print) | arXiv
[8] Morita equivalences of vector bundles, 2016 (e-print) | arXiv
[9] Grothendieck groups of Poisson vector bundles, J. Symplectic Geom., Volume 1 (2001), pp. 121-170 (e-print) | arXiv
[10] Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Adv. Math., Volume 223 (2010), pp. 1236-1275 (e-print) | arXiv
[11] A cohomology for foliated manifolds, Bull. Amer. Math. Soc., Volume 79 (1973), pp. 1283-1285
[12] P.P. La Pastina, L. Vitagliano, Deformations of linear Lie brackets, in preparation.
[13] On the strong homotopy Lie–Rinehart algebra of foliation, Commun. Contemp. Math., Volume 16 (2014) (e-print) | arXiv
[14] Morita equivalence of Poisson manifolds, Commun. Math. Phys., Volume 142 (1991), pp. 493-509
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