Filtrations on graph complexes and the Grothendieck–Teichmüller Lie algebra in depth two

• Matteo Felder ^[1]
  1. [1] Université de Genève

     Université de Genève

     Genève, Suiza

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 24, Nº. 3, 2018, págs. 2063-2092
• Idioma: inglés
• DOI: 10.1007/s00029-018-0416-0
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• Resumen

  • We establish an isomorphism between the Grothendieck–Teichmüller Lie algebra grt1 in depth two modulo higher depth and the cohomology of the two-loop part of the graph complex of internally connected graphs ICG(1) . In particular, we recover all linear relations satisfied by the brackets of the conjectural generators σ2k+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 GC2 via Willwacher’s isomorphism. We define a descending filtration on H0(GC2) and show that the degree two components of the corresponding associated graded vector spaces are isomorphic under Willwacher’s map.