On a lower central series filtration of the Grothendieck–Teichmüller Lie algebra $\mathfrak{grt}_1$

The Grothendieck–Teichmüller 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–Teichmüller Lie algebra, and we study the corresponding graded Lie algebra. Its degree zero part has been previously computed by the second author. We show that the degree one part is a module over a symmetric algebra such that both module and algebra are equipped with compatible decreasing filtrations. We exhibit an explicit lower bound for the associated graded module. We derive from that some information on the explicit expression of the depth 3 component of the associated graded Lie algebra (with respect to the depth filtration).