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 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) 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 grt 1 -elements implies the hexagon equation.Theorem 1.2. The automorphism of the Gerstenhaber operad up to homotopy are given by grt 1 and one class, i.e., there is an isomorphisms of Lie algebras H 0 (Der(hoe 2 )) ∼ = grt 1 ⋊ K =: grt where K acts on grt 1 by multiplication with the degree with respect to the grading on grt 1 .

Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields

“…We state the results of the third author from [44] which are used later in the text and introduce a couple of dg Lie algebras related to the full graph complex fGC.…”

“…This group is closely connected with the absolute Galois group Gal(Q / Q), it appears naturally in the study of moduli of algebraic curves, solutions of the Kashiwara-Vergne problem [1], theory of motives [6], [15], [26] and formal quantization procedures [18], [19], [43], [44]. The Lie algebra grt of GRT carries a natural grading by positive integers.…”

“…In paper [44], the third author established a link between the graph complex GC introduced in [38] by M. Kontsevich and the Lie algebra grt of the Grothendieck-Teichmüller group. More precisely, in [44], it was shown that H 0 (GC) ∼ = grt .…”

“…Using a link [44] between the graph complex GC and the deformation complex of the operad Ger, we prove that every cocycle γ ∈ GC gives us a derivation of the Gerstenhaber algebra (1.2).…”

Holonomies for connections with values in $L_\infty$-algebras

“…Proposition 5.17 (Proposition 6 from [33]). The map φ : t d (n) → H(CG d (n)), defined by sending t ij to the cohomology class of the graph that has only one edge going from the ith to the jth external vertices, is an isomorphism of graded Lie algebras.…”