On the compatibility between cup products, the Alekseev–Torossian connection and the Kashiwara–Vergne conjecture, II

Carlo A. Rossi

doi:10.1016/j.crma.2012.09.017

Lie Algebras/Mathematical Physics

On the compatibility between cup products, the Alekseev–Torossian connection and the Kashiwara–Vergne conjecture, II
[Compatibilité entre cup-produits, connexion dʼAlekseev–Torossian et conjecture de Kashiwara–Vergne, II]

Présenté par : Michèle Vergne

Carlo A. Rossi ¹

¹ MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany

Comptes Rendus. Mathématique, Volume 350 (2012) no. 19-20, pp. 871-874.

Résumé
Abstract

On donne ici une preuve différente de la compatibilité entre cup produits (Kontsevich, 2003, [3, Section 8]) en cohomologie de degré 0 dans le cas dʼune algèbre de Lie de dimension finie, dʼoù on déduit, en utilisant la connection de Alekseev–Torossian (Alekseev et Torossian, 2010, [1]), une écriture alternative du produit-étoilé de Kontsevich.

We give a different proof of the famous result on compatibility between cup product (Kontsevich, 2003, [3, Section 8]) in cohomology of degree 0, for a finite-dimensional Lie algebra, from which we deduce an alternative way of re-writing Kontsevichʼs star product by means of the Alekseev–Torossian connection (Alekseev and Torossian, 2010, [1]).

Reçu le : 2012-06-05
Accepté le : 2012-08-07
Publié le : 2012-10-01

DOI : 10.1016/j.crma.2012.09.017

Affiliations des auteurs :

Carlo A. Rossi ¹

¹ MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany

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     author = {Carlo A. Rossi},
     title = {On the compatibility between cup products, the {Alekseev{\textendash}Torossian} connection and the {Kashiwara{\textendash}Vergne} conjecture, {II}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {871--874},
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Carlo A. Rossi. On the compatibility between cup products, the Alekseev–Torossian connection and the Kashiwara–Vergne conjecture, II. Comptes Rendus. Mathématique, Volume 350 (2012) no. 19-20, pp. 871-874. doi : 10.1016/j.crma.2012.09.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.09.017/

Bibliographie
Cité par

[1] A. Alekseev; C. Torossian Kontsevich deformation quantization and flat connections, Comm. Math. Phys., Volume 300 (2010) no. 1, pp. 47-64 | DOI

[2] M. Andler; S. Sahi; C. Torossian Convolution of invariant distributions: proof of the Kashiwara–Vergne conjecture, Lett. Math. Phys., Volume 69 (2004), pp. 177-203 | DOI

[3] M. Kontsevich Deformation quantization of Poisson manifolds, Lett. Math. Phys., Volume 66 (2003) no. 3, pp. 157-216

[4] C.A. Rossi On the compatibility between cup products, the Alekseev–Torossian connection and the Kashiwara–Vergne conjecture, I, C. R. Acad. Sci. Paris, Sér. I, Volume 350 (2012) no. 17–18, pp. 823-826 | DOI

[5] B. Shoikhet Vanishing of the Kontsevich integrals of the wheels, Dijon (Lett. Math. Phys.), Volume 56 (2001) no. 2, pp. 141-149 | DOI

[6] C. Torossian Sur la conjecture combinatoire de Kashiwara–Vergne, J. Lie Theory, Volume 12 (2002) no. 2, pp. 597-616

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