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

• 成果类型: Article
• 署名作者: Willwacher, Thomas
• 署名单位: University of Zurich
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0528-x
• 发表日期: 2015
• 页码: 671-760
• 关键词: deformation quantization; formality theorem; double shuffle; associators; conjecture; operads; EQUATIONS; motives; discs

摘要

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