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

Abstract

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 \({{\mathrm{Lie}}}\pi _1(\mathsf {MTM})\) of the category of mixed Tate motives over \(\mathbb {Z}\) with a filtered and graded Lie algebra \(\mathfrak {u}\). This Lie algebra, and in particular \(\mathfrak {u}\), admits a representation as derivations of the free Lie algebra on two generators. In this paper we study the image \(\mathscr {E}\) of this representation of \(\mathfrak {u}\), starting from some results by Aaron Pollack, who determined all the relations in a certain filtered quotient of \(\mathscr {E}\), and gave several examples of relations in low weights in \(\mathscr {E}\) that are connected to period polynomials of cusp forms on \({{\mathrm{SL}}}_2(\mathbb {Z})\). 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 of moulds, to which we give a brief introduction. We prove two useful general theorems on moulds in the appendices.