Kashiwara-Vergne and dihedral bigraded Lie algebras in mould theory

On the elliptic Kashiwara-Vergne Lie algebra

We recall the definitions of two independently defined elliptic versions of the Kashiwara-Vergne Lie algebra $\frak{krv}$, namely the Lie algebra $\frak{krv}^{(1,1)}$ constructed by A.Alekseev, N.Kawazumi, Y.Kuno and F.Naef arising from the study of graded formality isomorphisms associated to topological fundamental groups of surfaces, and the Lie algebra $\frak{krv}_{ell}$ defined using mould theoretic techniques arising from multiple zeta theory by E.Raphael and L.Schneps, and show that they coincide.