Elliptic double shuffle, Grothendieck–Teichmüller and mould theory

In this article we define an elliptic double shuffle Lie algebra \(\scriptstyle {{\mathfrak {ds}}_{ell}}\) that generalizes the well-known double shuffle Lie algebra \(\scriptstyle {{\mathfrak {ds}}}\) to the elliptic situation. The double shuffle, or dimorphic, relations satisfied by elements of the Lie algebra \(\scriptstyle {{\mathfrak {ds}}}\) express two families of algebraic relations between multiple zeta values that conjecturally generate all relations. In analogy with this, elements of the elliptic double shuffle Lie algebra \(\scriptstyle {{\mathfrak {ds}}_{ell}}\) are Lie polynomials having a dimorphic property called \(\scriptstyle {\Delta }\)-bialternality that conjecturally describes the (dual of the) set of algebraic relations between elliptic multiple zeta values, which arise as coefficients of a certain elliptic generating series (constructed explicitly in Lochak et al. [15]) in On elliptic multiple zeta values 2016, in preparation) and closely related to the elliptic associator defined by Enriquez [10]. We show that one of Ecalle’s major results in mould theory can be reinterpreted as yielding the existence of an injective Lie algebra morphism \(\scriptstyle {{\mathfrak {ds}}\rightarrow {\mathfrak {ds}}_{ell}}\). Our main result is the compatibility of this map with the tangential-base-point section \(\scriptstyle {\mathrm{Lie}\,\pi _1(MTM)\rightarrow \mathrm{Lie}\,\pi _1(MEM)}\) constructed by Hain and Matsumoto [14] and with the section \(\scriptstyle {{\mathfrak {grt}}\rightarrow {\mathfrak {grt}}_{ell}}\) mapping the Grothendieck–Teichmüller Lie algebra \(\scriptstyle {{\mathfrak {grt}}}\) into the elliptic Grothendieck–Teichmüller Lie algebra \(\scriptstyle {{\mathfrak {grt}}_{ell}}\) constructed by Enriquez. This compatibility is expressed by the commutativity of the following diagram (excluding the dotted arrow, which is conjectural).