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

By Thomas Willwacher. 2010

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We show that the zeroth cohomology of M. Kontsevich’s graph complex is isomorphic to the Grothendieck–Teichmüller Lie algebra $$\mathfrak {{grt}}_1$$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 $$\mathfrak {{grt}}_1$$grt1, up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) $$E_n$$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 $$\mathfrak {{grt}}_1$$grt1-elements implies the hexagon equation.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/805cf8037ee307f7b4bfc6a3689ee771042f3ace