Moulds, Bimoulds, and some Lie algebras

Abstract

The Lie algebra of multiple zeta values is realized as the space ${\mathrm{ARI}}_{\underline{al}\ast \underline{il}}^{\mathrm{pol}}$ 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 ${\mathrm{BARI}}_{\underline{il},swap}^{\mathrm{pol}}$ 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 ${\mathrm{ARI}}_{\underline{al}\ast \underline{il}}^{\mathrm{pol}}$ embeds into ${\mathrm{BARI}}_{\underline{il},swap}^{\mathrm{pol}}$, with the $ari$ bracket translating directly into the $uri$ bracket. Finally, we examine the associated-depth graded of ${\mathrm{BARI}}_{\underline{il},swap}^{\mathrm{pol}}$, extending the known depth-graded setup for ${\mathrm{ARI}}_{\underline{al}\ast \underline{il}}^{\mathrm{pol}}$.

69 pages, comments are welcome!