M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra

Abstract

We show that the zeroth cohomology of M. Kontsevich’s graph complex is isomorphic to the Grothendieck–Teichmüller Lie algebra grt1. The map is explicitly described. This result has applications to deformation quantization and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber operad. They are parameterized by grt1, up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) En operads may be expressed through the cohomology of a suitable graph complex. Our methods also give a second proof of a result of H. Furusho, stating that the pentagon equation for grt1-elements implies the hexagon equation.