On the compatibility of the Betti harmonic coproduct with cyclotomic filtrations

TL;DR

This paper demonstrates that the Betti harmonic coproduct for cyclotomic filtrations remains compatible with the filtration structure, providing insights toward an explicit formula for the coproduct in the Betti double shuffle theory.

Contribution

It proves the compatibility of the classical Betti harmonic coproduct with cyclotomic filtrations for all N, advancing the understanding of the coproduct's explicit realization.

Findings

Compatibility of the Betti harmonic coproduct with cyclotomic filtrations.

Supports the conjecture that the completed coproduct can be explicitly realized.

Provides a foundation for future explicit formulas in Betti double shuffle theory.

Abstract

In a previous paper, the second author introduced a Betti counterpart of $N$-cyclotomic double shuffle theory for any $N \geq 1$. The construction is based on the group algebra of the free group $F_2$, endowed with a filtration relative to a morphism $F_2 \to \mu_N$ (where $\mu_N$ is the group of $N$-th roots of unity). One of the main results therein is the construction of a complete Hopf algebra coproduct $\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N$ on the relative completion of a specific subalgebra $\mathcal{W}^\mathrm{B}$ of the group algebra of $F_2$. However, an explicit formula for this coproduct is missing. In this paper, we show that the discrete Betti harmonic coproduct $\Delta^{\mathcal{W}, \mathrm{B}}$ defined in \cite{EF1} for the classical case ($N=1$) by the first author and Furusho remains compatible with the filtration structure on $\mathcal{W}^\mathrm{B}$ induced by the relative completion for arbitrary $N$. This compatibility suggests that the completion corresponding to $\Delta^{\mathcal{W}, \mathrm{B}}$ is a candidate for an explicit realization of $\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N$.