Relations dans l'alg\`ebre de Lie fondamentale des motifs elliptiques mixtes

TL;DR

This paper proves a conjecture about the existence of certain relations in the fundamental Lie algebra of universal mixed elliptic motives, extending Pollack's examples to all weights in depth 3.

Contribution

It establishes the existence of specific relations in the fundamental Lie algebra of mixed elliptic motives for all weights in depth 3, confirming a conjecture inspired by Pollack's examples.

Findings

Proves the conjecture in depth 3 for all weights

Establishes the existence of relations of Pollack's type in all weights

Extends understanding of the structure of the fundamental Lie algebra

Abstract

Hain and Matsumoto constructed a category of universal mixed elliptic motives and described the fundamental Lie algebra of this category, relating it to a certain graded and filtered Lie algebra E. In an unpublished paper, Aaron Pollack proved a result on relations in a certain quotient of E, and gave several examples of actual relations in small weight that naturally lead to a conjecture about the existence of such relations in all weights. In this article, we prove this conjecture in depth 3, establishing the existence in all weights of relations of the type noted in Pollack's examples. ----- Richard Hain et Makoto Matsumoto ont construit une cat\'egorie de motifs elliptiques mixtes universels et d\'ecrit l'alg\`ebre de Lie fondamentale de cette cat\'egorie en la reliant \`a une certaine alg\`ebre de Lie gradu\'ee et filtr\'ee E. Dans un article non publi\'e, Aaron Pollack a d\'emontr\'e un r\'esultat sur les relations dans un certain quotient de E, et donn\'e plusieurs exemples de relations en petit poids qui conduisent \`a une conjecture sur l'existence de telles relations en tout poids. Dans le pr\'esent article, nous prouvons cette conjecture en profondeur 3, \'etablissant l'existence en tout poids de relations du type d\'ecrit dans les exemples de Pollack.