Crossed product interpretation of the Double Shuffle Lie algebra attached to a finite Abelian group

Abstract

Racinet studied the scheme associated with the double shuffle and regularization relations between multiple polylogarithm values at $N^{th}$ roots of unity and constructed a group scheme attached to the situation; he also showed it to be the specialization for $G={\mu }_N$ of a group scheme ${\mathsf{DMR}}_0^G$ attached to a finite abelian group $G$. Then, Enriquez and Furusho proved that ${\mathsf{DMR}}_0^G$ can be essentially identified with the stabilizer of a coproduct element arising in Racinet's theory with respect to the action of a group of automorphisms of a free Lie algebra attached to $G$. We reformulate Racinet's construction in terms of crossed products. Racinet's coproduct can then be identified with a coproduct ${\hat{\Delta }}_G^{\mathcal{M}}$ defined on a module ${\hat{\mathcal{M}}}_G$ over an algebra ${\hat{\mathcal{W}}}_G$, which is equipped with its own coproduct ${\hat{\Delta }}_G^{\mathcal{W}}$, and the group action on ${\hat{\mathcal{M}}}_G$ extends to a compatible action of ${\hat{\mathcal{W}}}_G$. We then show that the stabilizer of ${\hat{\Delta }}_G^{\mathcal{M}}$, hence ${\mathsf{DMR}}_0^G$, is contained in the stabilizer of ${\hat{\Delta }}_G^{\mathcal{W}}$. This yields an explicit group scheme containing ${\mathsf{DMR}}_0^G$, which we also express in the Racinet formalism.