Comptes Rendus
Lie Algebras/Differential Geometry
Exponential map and L algebra associated to a Lie pair
[Application exponentielle et algébre L associée à une paire de Lie]
Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 817-821.

In this Note, we unveil homotopy-rich algebraic structures generated by the Atiyah classes relative to a Lie pair (L,A) of algebroids. In particular, we prove that the quotient L/A of such a pair admits an essentially canonical homotopy module structure over the Lie algebroid A, which we call Kapranov module.

Dans cette note, nous dévoilons des structures algébriques, riches en homotopies, engendrées par les classes dʼAtiyah relatives à une paire de Lie (L,A) dʼalgébroïdes. En particulier, nous prouvons que le quotient L/A dʼune telle paire admet une structure essentiellement canonique de module à homotopie près sur lʼalgébroïde de Lie A que nous appelons module de Kapranov.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.08.009

Camille Laurent-Gengoux 1 ; Mathieu Stiénon 2 ; Ping Xu 2

1 Département de mathématiques, université de Lorraine, île du Saulcy, 57000 Metz, France
2 Department of Mathematics, Penn State University, University Park, PA 16802, USA
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Camille Laurent-Gengoux; Mathieu Stiénon; Ping Xu. Exponential map and $ {L}_{\infty }$ algebra associated to a Lie pair. Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 817-821. doi : 10.1016/j.crma.2012.08.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.08.009/

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  • Zhuo Chen; Yu Qiao; Maosong Xiang; Tao Zhang Dg Loday–Pirashvili modules over Lie algebras, Journal of Homotopy and Related Structures, Volume 20 (2025) no. 1, p. 23 | DOI:10.1007/s40062-024-00361-6
  • Hsuan-Yi Liao Atiyah Classes and Todd Classes of Pullback dg Lie Algebroids Associated with Lie Pairs, Communications in Mathematical Physics, Volume 404 (2023) no. 2, p. 701 | DOI:10.1007/s00220-023-04854-y
  • Alexei Kotov; Camille Laurent-Gengoux; Vladimir Salnikov Normal forms of Z-graded Q-manifolds, Journal of Geometry and Physics, Volume 191 (2023), p. 104908 | DOI:10.1016/j.geomphys.2023.104908
  • Seokbong Seol; Mathieu Stiénon; Ping Xu Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras, Communications in Mathematical Physics, Volume 391 (2022) no. 1, p. 33 | DOI:10.1007/s00220-021-04265-x
  • Camille Laurent-Gengoux; Mathieu Stiénon; Ping Xu Poincaré–Birkhoff–Witt isomorphisms and Kapranov dg-manifolds, Advances in Mathematics, Volume 387 (2021), p. 107792 | DOI:10.1016/j.aim.2021.107792
  • Jiahao Cheng; Zhuo Chen; Dadi Ni Hopf algebras arising from dg manifolds, Journal of Algebra, Volume 584 (2021), p. 19 | DOI:10.1016/j.jalgebra.2021.05.004
  • Hsuan-Yi Liao; Mathieu Stiénon; Ping Xu Formality and Kontsevich–Duflo type theorems for Lie pairs, Advances in Mathematics, Volume 352 (2019), p. 406 | DOI:10.1016/j.aim.2019.04.047
  • Zhuo Chen; Honglei Lang; Maosong Xiang Atiyah Classes of Strongly Homotopy Lie Pairs, Algebra Colloquium, Volume 26 (2019) no. 02, p. 195 | DOI:10.1142/s1005386719000178
  • Zhuo Chen; Maosong Xiang; Ping Xu Atiyah and Todd classes arising from integrable distributions, Journal of Geometry and Physics, Volume 136 (2019), p. 52 | DOI:10.1016/j.geomphys.2018.10.011
  • Zhuo Chen; Mathieu Stiénon; Ping Xu From Atiyah Classes to Homotopy Leibniz Algebras, Communications in Mathematical Physics, Volume 341 (2016) no. 1, p. 309 | DOI:10.1007/s00220-015-2494-6
  • Luca Vitagliano On the strong homotopy associative algebra of a foliation, Communications in Contemporary Mathematics, Volume 17 (2015) no. 02, p. 1450026 | DOI:10.1142/s0219199714500266

Cité par 11 documents. Sources : Crossref

Research partially supported by the National Science Foundation [DMS-1101827] and the National Security Agency [H98230-12-1-0234].

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