On the combinatorics of the graph-complex

Bodo Lass

doi:10.1016/S1631-073X(02)02204-5

On the combinatorics of the graph-complex
[Sur la combinatoire du complexe de graphes]

Bodo Lass ^1,²

¹ Département de mathématique, Université Louis-Pasteur, 67084 Strasbourg, France
² Lehrstuhl II für Mathematik, RWTH Aachen, 52062 Aachen, Allemagne

Comptes Rendus. Mathématique, Volume 334 (2002) no. 1, pp. 1-6.

Résumé
Abstract

Dans leur prépublication récente [3], Kontsevich et Shoikhet ont introduit deux complexes de graphes : le complexe sur l'espace pair (resp. impair) pour étudier la cohomologie de l'algèbre de Lie Ham₀ (resp. Ham₀^odd) des champs vectoriels hamiltoniens sans terme constant sur l'espace pair (resp. impair) de dimension infinie. Nous construisons un isomorphisme entre ces deux complexes de graphes, démontrant notamment que leur cohomologies coïncident. Ceci résoud un problème posé par Shoikhet.

In their recent preprint [3] Kontsevich and Shoikhet have introduced two graph-complexes: the complex on the even (resp. odd) space in order to study the cohomology of the Lie algebra Ham₀ (resp. Ham₀^odd) of Hamiltonian vector fields vanishing at the origin on the infinite-dimensional even (resp. odd) space. We construct an isomorphism between those two graph-complexes, proving in particular that their cohomologies coincide. This solves a problem posed by Shoikhet.

Reçu le : 2001-10-26
Accepté le : 2001-11-13
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02204-5

Affiliations des auteurs :

Bodo Lass ^{1, 2}

¹ Département de mathématique, Université Louis-Pasteur, 67084 Strasbourg, France
² Lehrstuhl II für Mathematik, RWTH Aachen, 52062 Aachen, Allemagne

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Bodo Lass. On the combinatorics of the graph-complex. Comptes Rendus. Mathématique, Volume 334 (2002) no. 1, pp. 1-6. doi : 10.1016/S1631-073X(02)02204-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02204-5/

Bibliographie
Cité par

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[2] M. Kontsevich Formal (non)-commutative symplectic geometry (L. Corwin; I. Gelfand; J. Lepowsky, eds.), The Gelfand Mathematical Seminars 1990–1992, Birkhäuser, 1993, pp. 173-187

[3] Kontsevich M., Shoikhet B., Formality conjecture, geometry of complex manifolds and combinatorics of the graph-complex, Preprint

[4] Lass B., Une conjecture de Kontsevich et Shoikhet et la caractéristique d'Euler, Rev. Math. Pures Appl. (à paraı̂tre)

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