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

Carlo A. Rossi

doi:10.1016/j.crma.2012.08.002

Lie Algebras/Mathematical Physics

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

Présenté par : Michèle Vergne

Carlo A. Rossi ¹

¹ MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany

Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 823-826.

Résumé
Abstract

Soit $g$ une algèbre de Lie de dimension finie sur un champ $K \supset C$ . Du résultat (Kontsevich, 2003, [6, Section 8]) sur la compatibilité entre cup produits et de la conjecture de Kashiwara–Vergne (Kashiwara et Vergne, 1978, [4]), on déduit deux écritures alternatives du produit-étoilé de Kontsevich sur $S (g)$ .

Dans cette première partie, on fixe la notation et les conventions ; on rappelle le produit-étoilé de Kontsevich, on discute la conjecture de Kashiwara–Vergne et la relation entre elle et le produit-étoilé de Kontsevich.

For a finite-dimensional Lie algebra $g$ over a field $K \supset C$ , we deduce from the compatibility between cup products (Kontsevich, 2003, [6, Section 8]), on the one hand, and from the Kashiwara–Vergne conjecture (Kashiwara and Vergne, 1978, [4]), on the other hand, alternative ways of re-writing Kontsevich product ⋆ on $S (g)$ .

In this first part, we fix notation and conventions, revise the main features of Kontsevichʼs star product and examine the Kashiwara–Vergne conjecture and its relationship with Kontsevichʼs star product.

Reçu le : 2012-06-05
Accepté le : 2012-08-07
Publié le : 2012-09-01

DOI : 10.1016/j.crma.2012.08.002

Affiliations des auteurs :

Carlo A. Rossi ¹

¹ MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany

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     title = {On the compatibility between cup products, the {Alekseev{\textendash}Torossian} connection and the {Kashiwara{\textendash}Vergne} conjecture, {I}},
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Carlo A. Rossi. On the compatibility between cup products, the Alekseev–Torossian connection and the Kashiwara–Vergne conjecture, I. Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 823-826. doi : 10.1016/j.crma.2012.08.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.08.002/

Bibliographie
Cité par

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[2] J. Alm Two-colored noncommutative Gerstenhaber formality and infinity Duflo isomorphism, 2011 | arXiv

[3] 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

[4] M. Kashiwara; M. Vergne The Campbell–Hausdorff formula and invariant hyperfunctions, Invent. Math., Volume 47 (1978) no. 3, pp. 249-272

[5] V. Kathotia Kontsevichʼs universal formula for deformation quantization and the Campbell–Baker–Hausdorff formula, Internat. J. Math., Volume 11 (2000) no. 4, pp. 523-551 | DOI

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

[7] T. Mochizuki On the morphism of Duflo–Kirillov type, J. Geom. Phys., Volume 41 (2002) no. 1–2, pp. 73-113 | DOI

[8] C.A. Rossi The explicit equivalence between the standard and the logarithmic star product for Lie algebras, II, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012) no. 15–16, pp. 745-748

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

[10] D. Calaque; G. Felder; C.A. Rossi Deformation quantization with generators and relations, J. Algebra, Volume 337 (2011), pp. 1-12 | DOI

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

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