Abstract

We introduce the Chevalley cohomology for the graded Lie algebra of polyvector fields on R^d. This cohomology occurs naturally in the problem of construction and classification of formalities on the space R^d. Considering only graph formalities, that is, formalities defined with the help of graphs as in the original construction of Kontsevich, we define (as the first and third authors did earlier for the Hochschild cohomology) the Chevalley cohomology directly on spaces of graphs. More precisely, observing first a noteworthy property for Kontsevich’s explicit formality on R^d, 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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