Abstract
We establish an isomorphism between the Grothendieck–Teichmüller Lie algebra \(\mathfrak {grt}_1\) in depth two modulo higher depth and the cohomology of the two-loop part of the graph complex of internally connected graphs \(\mathsf {ICG}(1)\). In particular, we recover all linear relations satisfied by the brackets of the conjectural generators \(\sigma _{2k+1}\) modulo depth three by considering relations among two-loop graphs. The Grothendieck–Teichmüller Lie algebra is related to the zeroth cohomology of Kontsevich’s graph complex \(\mathsf {GC}_2\) via Willwacher’s isomorphism. We define a descending filtration on \(H^0(\mathsf {GC}_2)\) and show that the degree two components of the corresponding associated graded vector spaces are isomorphic under Willwacher’s map.
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Felder, M. Filtrations on graph complexes and the Grothendieck–Teichmüller Lie algebra in depth two. Sel. Math. New Ser. 24, 2063–2092 (2018). https://doi.org/10.1007/s00029-018-0416-0
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DOI: https://doi.org/10.1007/s00029-018-0416-0