On a lower central series filtration of the Grothendieck-Teichm\"uller Lie algebra grt_1

TL;DR

This paper investigates a filtration of the Grothendieck-Teichm"uller Lie algebra induced by the lower central series of a larger derivation Lie algebra, providing new insights into its graded structure and depth components.

Contribution

It introduces a new filtration on the Grothendieck-Teichm"uller Lie algebra and analyzes its graded Lie algebra, including explicit bounds for the depth-graded parts.

Findings

Degree zero part previously computed

Degree one part is a module over a symmetric algebra

Provides explicit lower bounds for the graded module

Abstract

The Grothendieck-Teichm\"uller Lie algebra is a Lie subalgebra of a Lie algebra of derivations of the free Lie algebra in two generators. We show that the lower central series of the latter Lie algebra induces a decreasing filtration of the Grothendieck-Teichm\"uller Lie algebra and we study the corresponding graded Lie algebra. Its degree zero part had been previously computed by the second author. We show that the degree one part is a module over a symmetric algebra, which are both equipped with compatible decreasing filtrations, and we exhibit an explicit lower bound for the associated graded module. We derive from there some information on explicit expression of the depth 3 part of the depth-graded of the Grothendieck-Teichm\"uller Lie algebra.