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

2010/09/30 by Thomas Willwacher · 1 citation
Mathematics · #math.QA

paper · pdf · doi:10.1007/s00222-014-0528-x

arxiv created 2013/12/23 · arxiv updated 2015/02/23

Abstract

We show that the zeroth cohomology of M. Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmueller 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) 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 grt₁-elements implies the hexagon equation.

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• Tamarkin's construction is equivariant with respect to the action of the Grothendieck-Teichmueller group