Moulds, Bimoulds, and some Lie algebras

Source abstract

The Lie algebra of multiple zeta values is realized as the space $\operatorname{ARI}^{\operatorname{pol}}_{\underline{al}\ast \underline{il}}$ of alternal moulds whose swap is alternil up to a constant mould, equipped with the $ari$ bracket. In this paper, we study the larger space $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$ of alternil, swap-invariant bimoulds, which is conjecturally the Lie algebra for multiple $q$-zeta values. We propose an explicit formula for a Lie bracket $uri$ on this space. Moreover, we prove that the Lie algebra $\operatorname{ARI}^{\operatorname{pol}}_{\underline{al}\ast \underline{il}}$ embeds into $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$, with the $ari$ bracket translating directly into the $uri$ bracket. Finally, we examine the associated-depth graded of $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$, extending the known depth-graded setup for $\operatorname{ARI}_{\underline{al}\ast \underline{il}}^{\operatorname{pol}}$.