On the derivation representation of the fundamental Lie algebra of mixed elliptic motives

Richard Hain and Makoto Matsumoto constructed a category of universal mixed elliptic motives, and described the fundamental Lie algebra of this category: it is a semi-direct product of the fundamental Lie algebra Lie${\pi }_1(MTM)$ of the category of mixed Tate motives over $\mathbf{Z}$ with a filtered and graded Lie algebra $u$. This Lie algebra, and in particular the subspace $u$, admits a representation as derivations of the free Lie algebra on two generators. In this paper we study the image $E$ of this representation of $u$, starting from some results by Aaron Pollack, who determined all the relations in a certain filtered quotient of $E$, and gave several examples of relations in low weights in $E$ that are connected to period polynomials of cusp forms on $SL_2({\bf Z})$. Pollack’s examples lead to a conjecture on the existence of such relations in all depths and all weights, that we state in this article and prove in depth 3 in all weights. The proof follows quite naturally from Ecalle’s theory moulds, to which we give a brief introduction. We prove two useful general theorems on moulds in the appendices.