Filtrations on graph complexes and the Grothendieck-Teichm "uller Lie\n algebra in depth two

2017/07/03 by Matteo Felder, Felder, Matteo
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.1707.00495

openalex publication_date 2017/07/03 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

We establish an isomorphism between the Grothendieck-Teichm "uller Lie\nalgebra mathfrakgrt₁ in depth two modulo higher depth and the cohomology\nof the two-loop part of the graph complex of internally connected graphs\n\ICG(1). In particular, we recover all linear relations satisfied by\nthe brackets of the conjectural generators \σ_2k+1 modulo depth three\nby considering relations among two-loop graphs.\n The Grothendieck-Teichm "uller Lie algebra is related to the zeroth\ncohomology of M. Kontsevich's graph complex \GC₂ via T. Willwacher's\nisomorphism. We define a descending filtration on H⁰(\GC₂) and show\nthat the degree two components of the corresponding associated graded vector\nspaces are isomorphic under T. Willwacher's map.\n

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