M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra

TL;DR

This paper establishes an isomorphism between Kontsevich's graph complex cohomology and the Grothendieck-Teichmueller Lie algebra, with implications for deformation quantization and operad theory.

Contribution

It explicitly describes the isomorphism between the graph complex cohomology and grt_1, and connects this to derivations of operads and the pentagon-hexagon equations.

Findings

Zeroth cohomology of graph complex is isomorphic to grt_1.

Homotopy derivations of Gerstenhaber operad are parameterized by grt_1.

Provides a new proof relating pentagon and hexagon equations.

Abstract

We show that the zeroth cohomology of M. Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmueller Lie algebra grt_1. 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 grt_1, up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) E_n 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 grt_1-elements implies the hexagon equation.