A lower bound of a sub-quotient of the Lie algebra associated to Grothendieck-Teichmüller group
Seminar Room 2, Newton Institute Gatehouse
AbstractWe show that the filtration given by the central descending series of the commutator of the free Lie algebra on two generators x,y induces by a filtration of the graded Lie algebra grt_1 associated to the Grothendieck-Teichmüller group. The degree 0 part of the associated graded space has already been computed (by the collaborator of the author). We get here a lower bound for the degree 1 part; more precisely, this graded space splits into a sum of homogeneous components, on which we get a filtration and we give a lower bound for the dimensions of each sub-quotient.
The proof uses the construction of a vector space included in certain Lie sub-algebras of extensions between abelian Lie algebras, and reduces the problem to a question of commutative algebras, which is treated with invariant theory and results of Ihara, Takao, and Schneps on the quadratic relations between elements of the degree 1 part associated to grt_1 for the depth filtration (corresponding to the y-degree). As a corollary, we give another proof of a statement of Ecalle describing the sub-space of the degree 2 part of the same graded space.
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