Abstract
We introduce the Chevalley cohomology for the graded Lie algebra of polyvector ?elds on Rd. This cohomology occurs naturally in the problem of construction and classi?cation of formalities on the space Rd. Considering only graph formalities, that is, formalities de?ned with the help of graphs as in the original construction of Kontsevich, we de?ne (as the ?rst and third authors did earlier for the Hochschild cohomology) the Chevalley cohomology directly on spaces of graphs. More precisely, observing ?rst a noteworthy property for Kontsevich?s explicit formality on Rd, we restrict ourselves to graph formalities with that property. With this restriction, we obtain some simple expressions for the Chevalley coboundary operator; in particular, we can write this cohomology directly on the space of purely aerial, nonoriented graphs. We also give examples and applications.
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