The cyclotomic double shuffle torsor in terms of Betti and de Rham coproducts

Abstract

To describe the double shuffle relations between multiple polylogarithm values at $N$th roots of unity, Racinet attached to each finite cyclic group $G$ of order $N$ and each group embedding $\iota :G\to {\mathbb{C}}^{\times }$, a $\mathbb{Q}$-scheme ${\mathsf{DMR}}^{\iota }$ which associates to each commutative $\mathbb{Q}$-algebra $\mathbf{k}$, a set ${\mathsf{DMR}}^{\iota }(\mathbf{k})$ that can be decomposed as a disjoint union of sets ${\mathsf{DMR}}_{\lambda }^{\iota }(\mathbf{k})$ with $\lambda \in \mathbf{k}$. He also exhibited a $\mathbb{Q}$-group scheme ${\mathsf{DMR}}_0^G$ and showed that ${\mathsf{DMR}}_{\lambda }^{\iota }(\mathbf{k})$ is a torsor for the action of ${\mathsf{DMR}}_0^G(\mathbf{k})$. Then, Enriquez and Furusho showed for $N=1$ that a subscheme ${\mathsf{DMR}}_{\times }^{\iota }$ of ${\mathsf{DMR}}^{\iota }$ is a torsor of isomorphisms relating de Rham and Betti objects. In previous work, we reformulated Racinet's construction in terms of crossed products and identified his coproduct with a coproduct ${\hat{\Delta }}_G^{\mathcal{M},\mathrm{DR}}$ defined on a module ${\hat{\mathcal{M}}}_G^{\mathrm{DR}}$ over an algebra ${\hat{\mathcal{W}}}_G^{\mathrm{DR}}$ equipped with its own coproduct ${\hat{\Delta }}_G^{\mathcal{W},\mathrm{DR}}$. In this paper, we provide a generalization of Enriquez and Furusho's result to any $N\ge 1$: we exhibit a module ${\hat{\mathcal{M}}}_N^{\mathrm{B}}$ over an algebra ${\hat{\mathcal{W}}}_N^{\mathrm{B}}$ and show the existence of compatible coproducts ${\hat{\Delta }}_N^{\mathcal{W},\mathrm{B}}$ and ${\hat{\Delta }}_N^{\mathcal{M},\mathrm{B}}$ such that ${\mathsf{DMR}}_{\times }^{\iota }$ is contained in the torsor of isomorphisms relating ${\hat{\Delta }}_N^{\mathcal{W},\mathrm{B}}$ (resp. ${\hat{\Delta }}_N^{\mathcal{M},\mathrm{B}}$) to ${\hat{\Delta }}_G^{\mathcal{W},\mathrm{DR}}$ (resp. ${\hat{\Delta }}_G^{\mathcal{M},\mathrm{DR}}$).