A Hopf algebra associated with a Lie pair

Zhuo Chen; Mathieu Stiénon; Ping Xu

doi:10.1016/j.crma.2014.09.010

Differential geometry

A Hopf algebra associated with a Lie pair
[Une algèbre de Hopf associée à une paire de Lie]

Présenté par : Charles-Michel Marle

Zhuo Chen ¹ ; Mathieu Stiénon ² ; Ping Xu ²

¹ Department of Mathematics, Tsinghua University, China
² Department of Mathematics, Penn State University, United States

Comptes Rendus. Mathématique, Volume 352 (2014) no. 11, pp. 929-933.

Abstract (VO)
Résumé

The quotient $L / A [- 1]$ of a pair $A ↪ L$ of Lie algebroids is a Lie algebra object in the derived category $D^{b} (A)$ of the category $A$ of left $U (A)$ -modules, the Atiyah class $α_{L / A}$ being its Lie bracket. In this note, we describe the universal enveloping algebra of the Lie algebra object $L / A [- 1]$ and we prove that it is a Hopf algebra object in $D^{b} (A)$ .

Le quotient $L / A [- 1]$ d'une paire $A ↪ L$ d'algébroïdes de Lie est un objet algèbre de Lie dans la catégorie dérivée $D^{b} (A)$ de la catégorie $A$ des modules à gauche sur $U (A)$ . Dans cette note, nous décrivons l'algèbre enveloppante universelle de l'objet algèbre de Lie $L / A [- 1]$ et nous prouvons que celle-ci constitue un objet algèbre de Hopf dans $D^{b} (A)$ .

Reçu le : 2013-08-06
Accepté le : 2014-09-16
Publié le : 2014-10-03

DOI : 10.1016/j.crma.2014.09.010

Affiliations des auteurs :

Zhuo Chen ¹ ; Mathieu Stiénon ² ; Ping Xu ²

¹ Department of Mathematics, Tsinghua University, China
² Department of Mathematics, Penn State University, United States

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Zhuo Chen; Mathieu Stiénon; Ping Xu. A Hopf algebra associated with a Lie pair. Comptes Rendus. Mathématique, Volume 352 (2014) no. 11, pp. 929-933. doi : 10.1016/j.crma.2014.09.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.09.010/

Bibliographie
Cité par

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Zhuo Chen; Maosong Xiang; Ping Xu Hochschild cohomology of dg manifolds associated to integrable distributions, Communications in Mathematical Physics, Volume 396 (2022) no. 2, pp. 647-684 | DOI:10.1007/s00220-022-04473-z | Zbl:1510.58002
Jiahao Cheng; Zhuo Chen; Dadi Ni Hopf algebras arising from dg manifolds, Journal of Algebra, Volume 584 (2021), pp. 19-68 | DOI:10.1016/j.jalgebra.2021.05.004 | Zbl:1475.58003
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; Mathieu Stiénon; Ping Xu From Atiyah classes to homotopy Leibniz algebras, Communications in Mathematical Physics, Volume 341 (2016) no. 1, pp. 309-349 | DOI:10.1007/s00220-015-2494-6 | Zbl:1395.53090

Cité par 4 documents. Sources : Crossref, zbMATH

^☆ Research partially supported by NSF grant DMS1101827, NSA grant H98230-12-1-0234, and NSFC grants 11001146 and 11471179.

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