Dual-depth Adapted Irreducible Formal Multizeta Values

Abstract

Let $\mathfrak{ds}$ denote the double shuffle Lie algebra, equipped with the standard weight grading and depth filtration; we write $\mathfrak{ds}=\oplus_{n\ge 3} \mathfrak{ds}_n$ and denote the filtration by $\mathfrak{ds}^1\supset \mathfrak{ds}^2\supset \cdots$. The double shuffle Lie algebra is dual to the new formal multizeta space $\mathfrak{nfz}=\oplus_{n\ge 3} \mathfrak{nfz}_n$, which is equipped with the dual depth filtration $\mathfrak{nfz}^1\subset \mathfrak{nfz}^2\subset\cdots$ Via an explicit canonical isomorphism $\mathfrak{ds}\buildrel \sim\over\rightarrow\mathfrak{nfz}$, we define the "dual" in $\mathfrak{nfz}$ of an element in $\mathfrak{ds}$. For each weight $n\ge 3$ and depth $d\ge 1$, we then define the vector subspace $\mathfrak{ds}_{n,d}$ of $\mathfrak{ds}$ as the space of elements in $\mathfrak{ds}_n^d-\mathfrak{ds}_n^{d+1}$ whose duals lie in $\mathfrak{nfz}_n^d$. We prove the direct sum decomposition $$\mathfrak{ds}=\bigoplus_{n\ge 3}\bigoplus_{d\ge 1} \mathfrak{ds}_{n,d},$$ \noindent which yields a canonical vector space isomorphism between $\mathfrak{ds}$ and its associated graded for the depth filtration, $\mathfrak{ds}_{n,d}\simeq \mathfrak{ds}_n^d/ \mathfrak{ds}_n^{d+1}$. A basis of $\mathfrak{ds}$ respecting this decomposition is dual-depth adapted, which means that it is adapted to the depth filtration on $\mathfrak{ds}$, and the basis of dual elements is adapted to the dual depth filtration on $\mathfrak{nfz}$. We use this notion to give a canonical depth 1 generator $f_n$ for $\mathfrak{ds}$ in each odd weight $n\ge 3$, namely the dual of the new formal single zeta value $\zeta(n)\in\mathfrak{nfz}_n$. At the end, we also apply the result to give canonical irreducibles for the formal multizeta algebra in weights up to 12.