A lower bound of a sub-quotient of the Lie algebra associated to Grothendieck-Teichmüller group

Duration: 1 hour 29 mins

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Description:	Enriquez, B (University of Strasbourg) Thursday 18 April 2013, 16:00-18:00


Created:	2013-05-13 17:21
Collection:	Grothendieck-Teichmüller Groups, Deformation and Operads
Publisher:	Isaac Newton Institute
Copyright:	Enriquez, B
Language:	eng (English)


Abstract:	We 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.

Abstract:

We 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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