Elliptic multiple zeta values, Grothendieck-Teichm\"uller and mould theory

Abstract

In this article we define an elliptic double shuffle Lie algebra $ds_{ell}$ that generalizes the well-known double shuffle Lie algebra $ds$ to the elliptic situation. The double shuffle, or dimorphic, relations satisfied by elements of the Lie algebra $ds$ 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 $ds_{ell}$ are Lie polynomials having a dimorphic property called $\Delta$-bialternality that conjecturally describes the (dual of the) set of algebraic relations between elliptic multiple zeta values, periods of objects of the category $MEM$ of mixed elliptic motives defined by Hain and Matsumoto. 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 $ds\rightarrow ds_{ell}$. Our main result is the compatibility of this map with the tangential-base-point section ${\rm Lie}\,\pi_1(MTM)\rightarrow {\rm Lie}\,\pi_1(MEM)$ constructed by Hain and Matsumoto and with the section $grt\rightarrow grt_{ell}$ mapping the Grothendieck-Teichm\"uller Lie algebra $grt$ into the elliptic Grothendieck-Teichm\"uller Lie algebra $grt_{ell}$ constructed by Enriquez.