We introduce the
Chevalley cohomology for the graded Lie algebra of polyvector fields on ℝd. This
cohomology occurs naturally in the problem of construction and classification of
formalities on the space ℝd. 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 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.
