On a lower central series filtration of the Grothendieck-Teichmüller Lie algebra grt_1

Zeta Elements in Depth 3 and the Fundamental Lie Algebra of The infinitesimal Tate Curve

“…This is a finite computation and only uses the fact that x 2n 1 is an even function. It also follows from the fact that (s (1) , s (2) ), and x 2n 1 are solutions to the double shuffle equations via a version of Racinet's theorem for rational functions.…”

“…It is easy to verify that (s (1) , s (2) ) is a solution to the double shuffle equations (4.20) and (4.21) in depths one and two. = {s (1) , x 2n 1 } ξ…”

“…We shall only require their restriction to depths ≤ 3 and work with translation-invariant representations (4.3). Let (f (1) , f (2) , f (3)…”

“…This is discussed in [2], and implies the previous proposition. The component of s in depth 3 is unique by an extension of the depth-parity theorem for double shuffle equations (theorem 2.5) to the case of rational functions and given by s (3) = 1 2 {s (1) , s (2) } . 5.2.…”

“…We shall show as a consequence of [3] that (1.6) is injective. 1 If one thinks of g m as being bigraded for M and W , with W = M , then the map i 1 respects the M -grading, but not the W -grading, see [27].…”