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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Alekseev, A., Torossian, C.: The Kashiwara–Vergne conjecture and Drinfeld’s associators. Ann. Math. 172(2), 415–463 (2012)
Arone, G., Turchin, V.: Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots. Ann. Inst. Fourier (Grenoble) 65(1), 1–62 (2015)
Brown, F.: Mixed tate motives over \({\mathbb{Z}}\). Ann. Math. 175(2), 949–976 (2012)
Conant, J., Gerlits, F., Vogtmann, K.: Cut vertices in commutative graphs. Q. J. Math. 56(3), 321–336 (2005)
Conant, J., Costello, J., Turchin, V., Weed, P.: Two-loop part of the rational homotopy of spaces of long embeddings. J. Knot Theory Ramif. 23(4), 1450018 (2014)
Drinfeld, V.G.: On quasitriangular quasi-Hopf algebras and on agroup that is closely connected with \(({\bar{Q}}/\text{ Q })\). Leningr. Math. J. 2(4), 829–860 (1991)
Felder, M.: Internally connected graphs and the Kashiwara–Vergne Lie algebra. Lett. Math. Phys. 108(6), 1407–1441 (2018)
Goncharov, A.: The dihedral Lie algebras and Galois symmetries of \(\pi _1^{(l)}({\mathbb{P}}^1-(\{0,\infty \}\cup \mu _N))\). Duke Math. J. 110(3), 397–487 (2001)
Ihara, Y.: Some arithmetic aspects of Galois actions in the pro-\(p\) fundamental group of \({\mathbb{P}}^1-\{0,1,\infty \}\). In: Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999), volume 70 of Proc. Sympos. Pure Math., pages 247–273. Amer. Math. Soc., Providence, RI (2002)
Kontsevich, M.: Formal (non)commutative symplectic geometry. The Gel\(\prime \)fand Mathematical Seminars. 1990–1992, pp. 173–187. Birkhäuser Boston, Boston, MA (1993)
Kontsevich M.: Feynman diagrams and low-dimensional topology. In: First European Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages 97–121. Birkhäuser, Basel (1994)
Kontsevich, M.: Formality conjecture. In: Deformation theory and symplectic geometry (Ascona, 1996), volume 20 of Math. Phys. Stud., pages 139–156. Kluwer Acad. Publ., Dordrecht (1997)
Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48(1), 35–72 (1999)
Lambrechts, P., Volić, I.: Formality of the little \(N\)-disks operad. Mem. Am. Math. Soc. 230(1079), viii+116 (2014)
Schneps, L.: On the Poisson bracket on the free Lie algebra in two generators. J. Lie Theory 16(1), 19–37 (2006)
Ŝevera, P., Willwacher, T.: Equivalence of formalities of the little discs operad. Duke Math. J. 160(1), 175–206 (2011)
Willwacher, T.: M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra. Invent. Math. 200(3), 671–760 (2014)
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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