Lie Algebras/Differential Geometry
Exponential map and L algebra associated to a Lie pair
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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.08.009
Laurent-Gengoux, Camille 1; Stiénon, Mathieu 2; Xu, Ping 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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Laurent-Gengoux, Camille; Stiénon, Mathieu; Xu, Ping. 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. http://archive.numdam.org/articles/10.1016/j.crma.2012.08.009/

[1] Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C. Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys., Volume 165 (1994) no. 2, pp. 311-427 MR 1301851 (95f:32029)

[2] Calabi, Eugenio Isometric imbedding of complex manifolds, Ann. of Math. (2), Volume 58 (1953), pp. 1-23 MR 0057000 (15,160c)

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

[4] Costello, Kevin J. A geometric construction of the Witten genus, II, 2011 | arXiv

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

[6] Laurent-Gengoux, Camille; Stiénon, Mathieu; Xu, Ping Holomorphic Poisson manifolds and holomorphic Lie algebroids, Int. Math. Res. Not. IMRN (2008) Art. ID rnn 088, 46. MR 2439547 (2009i:53082)

[7] Nistor, Victor; Weinstein, Alan; Xu, Ping Pseudodifferential operators on differential groupoids, Pacific J. Math., Volume 189 (1999) no. 1, pp. 117-152 MR 1687747 (2000c:58036)

[8] Weinstein, Alan The integration problem for complex Lie algebroids, From Geometry to Quantum Mechanics, Progr. Math., vol. 252, Birkhäuser Boston, Boston, MA, 2007, pp. 93-109 (MR 2285039)

[9] Shilin Yu, Dolbeault dga of formal neighborhoods and L algebroids, Ph.D. thesis, Penn State University, State College, PA, 2013.

Cited by Sources:

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