no. 17-18
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

Rossi, Carlo A.1
1 MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany
Comptes Rendus. Mathématique, Tome 350 (2012) no. 17-18, pp. 823-826.

Soit g une algèbre de Lie de dimension finie sur un champ K⊃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⊃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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.08.002
Rossi, Carlo A. 1

1 MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany 
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Rossi, Carlo A. On the compatibility between cup products, the Alekseev–Torossian connection and the Kashiwara–Vergne conjecture, I. Comptes Rendus. Mathématique, Tome 350 (2012) no. 17-18, pp. 823-826. doi : 10.1016/j.crma.2012.08.002. http://www.numdam.org/articles/10.1016/j.crma.2012.08.002/

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

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

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

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

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

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

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

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

[11] Shoikhet, B. 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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