Alg\`ebres et cog\`ebres de Gerstenhaber et cohomologie de Chevalley-Harrison

The fundamental example of Gerstenhaber algebra is the space $T_{poly}({\mathbb{R}}^d)$ of polyvector fields on ${\mathbb{R}}^d$, equipped with the wedge product and the Schouten bracket. In this paper, we explicitely describe what is the enveloping $G_{\mathrm{\infty }}$ algebra of a Gerstenhaber algebra $\mathcal{G}$. This structure gives us a definition of the Chevalley-Harrison cohomology operator for $\mathcal{G}$. We finally show the nontriviality of a Chevalley-Harrison cohomology group for a natural Gerstenhaber subalgebra in $T_{poly}({\mathbb{R}}^d)$.