Homology, Homotopy and Applications

Volume 15 (2013)

Number 1

Chevalley cohomology for aerial Kontsevich graphs

Pages: 83 – 100

DOI: https://dx.doi.org/10.4310/HHA.2013.v15.n1.a5

Authors

Walid Aloulou (Laboratoire de Mathématique Physique Fonctions Spéciales et Applications, Université de Sousse; Département de Mathématiques, Institut Préparatoire aux Etudes d’Ingénieurs de Sfax, Sfax, Tunisia)

Didier Arnal (Institut de Mathématiques de Bourgogne, Université de Bourgogne, U.F.R. Sciences et Techniques, Dijon, France)

Ridha Chatbouri (Laboratoire de Mathématique Physique Fonctions Spéciales et Applications, Université de Sousse, France; Département de Mathématiques, Faculté des Sciences de Monastir, Tunisia)

Abstract

Let $T_{\operatorname{poly}}(\mathbb{R}^d)$ denote the space of skew-symmetric polyvector fields on $\mathbb{R}^d$, turned into a graded Lie algebra by means of the Schouten bracket. Our aim is to explore the cohomology of this Lie algebra, with coefficients in the adjoint representation, arising from cochains defined by linear combination of aerial Kontsevich graphs. We prove that this cohomology is localized at the space of graphs without any isolated vertex, any “hand” or any “foot”. As an application, we explicitly compute the cohomology of the “ascending graphs” quotient complex.

Keywords

Kontsevich graphs, Chevalley cohomology

2010 Mathematics Subject Classification

05C90, 17B56, 53D50

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Published 1 May 2013