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

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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 Lieπ₁(MTM) of the category of mixed Tate motives over 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 SL₂(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.