Chevalley cohomology for aerial Kontsevich graphs

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.