Cohomologie de Chevalley des graphes vectoriels

The space of smotth functions and vector fields on \(\R^d\) is a Lie subalgebra of the (graded) Lie algebra \(T\_{poly}(\R^d)\), equipped with the Scouten bracket. In this paper, we compute the cohomology of this subalgebra for the adjoint representation in \(T\_{poly}(\R^d)\), restricting ourselves to the case of cochains defined with purely aerial Kontsevich's graphs, as in [AGM]. We find results which are very similar to the classical Gelfand-Fuchs and de Wilde-Lecomte one.