Moulds, Bimoulds, 与若干李代数
Moulds, Bimoulds, and some Lie algebras
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中文总结 AI 辅助
本文研究多重zeta值李代数及其推广，提出alternil且交换不变bimoulds空间上的显式李括号uri，证明原李代数嵌入并考察深度分次，为多重q-zeta值李代数提供基础。
中文摘要 AI 辅助
多重zeta值的李代数被实现为空间 $\operatorname{ARI}^{\operatorname{pol}}_{\underline{al}\ast \underline{il}}$，该空间由alternal moulds（其交换（swap）在相差一个常数mould的意义下是alternil）构成，并配备 $ari$ 括号。本文研究了更大的空间 $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$，它由alternil且交换不变的bimoulds组成，该空间被猜想为多重 $q$-zeta值的李代数。我们提出了该空间上一个李括号 $uri$ 的显式公式。此外，我们证明了李代数 $\operatorname{ARI}^{\operatorname{pol}}_{\underline{al}\ast \underline{il}}$ 嵌入到 $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$ 中，且 $ari$ 括号直接转化为 $uri$ 括号。最后，我们考察了 $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$ 的关联深度分次，推广了 $\operatorname{ARI}_{\underline{al}\ast \underline{il}}^{\operatorname{pol}}$ 的已知深度分次结构。
英文摘要
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 $\operatorname{ARI}_{\underline{al}\ast \underline{il}}^{\operatorname{pol}}$.
