A Hopf algebra associated with a Lie pair

Chen, Zhuo; Stiénon, Mathieu; Xu, Ping

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

Chen, Zhuo ¹ ; Stiénon, Mathieu ² ; Xu, Ping ²

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

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

Résumé
Abstract

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 $D^{b} (A)$ .

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 $D^{b} (A)$ .

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 :

Chen, Zhuo ¹ ; Stiénon, Mathieu ² ; Xu, Ping ²

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

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Chen, Zhuo; Stiénon, Mathieu; Xu, Ping. A Hopf algebra associated with a Lie pair. Comptes Rendus. Mathématique, Tome 352 (2014) no. 11, pp. 929-933. doi : 10.1016/j.crma.2014.09.010. http://www.numdam.org/articles/10.1016/j.crma.2014.09.010/

Bibliographie
Cité par

[1] Chen, Zhuo; Stiénon, Mathieu; Xu, Ping From Atiyah classes to homotopy Leibniz algebras, 2012 | arXiv

[2] Kapranov, M. Rozansky–Witten invariants via Atiyah classes, Compos. Math., Volume 115 (1999) no. 1, pp. 71-113 MR 1671737 (2000h:57056)

[3] Kowalzig, Niels; Posthuma, Hessel The cyclic theory of Hopf algebroids, J. Noncommut. Geom., Volume 5 (2011) no. 3, pp. 423-476 MR 2817646 (2012f:16081)

[4] Liu, Zhang-Ju; Weinstein, Alan; Xu, Ping Dirac structures and Poisson homogeneous spaces, Commun. Math. Phys., Volume 192 (1998) no. 1, pp. 121-144 MR 1612164 (99g:58053)

[5] Markarian, Nikita The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem, J. Lond. Math. Soc. (2), Volume 79 (2009) no. 1, pp. 129-143 MR 2472137 (2010d:14020)

[6] Ramadoss, Ajay C. The big Chern classes and the Chern character, Int. J. Math., Volume 19 (2008) no. 6, pp. 699-746 MR 2431634 (2010h:14028)

[7] Roberts, Justin; Willerton, Simon On the Rozansky–Witten weight systems, Algebr. Geom. Topol., Volume 10 (2010) no. 3, pp. 1455-1519 (MR 2661534)

[8] Rozansky, L.; Witten, E. Hyper–Kähler geometry and invariants of three-manifolds, Sel. Math. New Ser., Volume 3 (1997) no. 3, pp. 401-458 MR 1481135 (98m:57041)

[9] Xu, Ping Quantum groupoids, Commun. Math. Phys., Volume 216 (2001) no. 3, pp. 539-581 MR 1815717 (2002f:17033)

Cité par Sources :

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