Divergence and \(q\)-divergence in depth 2 (Q2791853)

The Kashiwara-Vergne problem in Lie theory states the existence of an automorphism \(F\) of a \(2\)-dimensional Lie algebra which satisfies certain properties. At this respect, it is already known that the Kashiwara-Vergne Lie algebra \(\mathfrak{lrv}\) encodes symmetries of the Kashiwara-Vergne problem on the properties of the Campbell-Hausdorff series and it is conjectured that \(\mathfrak{lrv} \cong \mathbb{K}t\oplus\mathfrak{grt}_1\), where \(t\) is a generator of degree 1 and \(\mathfrak{grt}_1\) is the Grothendieck-Teichmüller Lie algebra. In this paper, authors deal with that conjecture and they prove it in depth 2 by using, as the main tools, the divergence cocycle and the representation theory of the dihedral group \(D_{12}\). Their calculation is similar to the one by Zagier of the graded dimensions of the double shuffle Lie algebra in depth 2. In analogy to the divergence cocycle, they define the super-divergence and \(q\)-divergence cocycles for \(q\) a primitive root of unity of order \(l\), \(q^l =1\), on Lie subalgebras of \(\mathfrak{grt}_1\) which consist of elements with weight divisible by \(l\). They also show that in depth 2 these cocycles have no kernel, which is in sharp contrast with the fact that the divergence cocycle vanishes on \([\mathfrak{grt}_1,\mathfrak{grt}_1]\). Finally, they conjecture that super-divergence and \(q\)-divergence cocycles have no kernel in arbitrary depth.