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

TL;DR

This paper provides a new interpretation of the Double Shuffle Lie algebra associated with a finite abelian group using crossed product structures, offering explicit descriptions of related group schemes and their stabilizers.

Contribution

It reformulates Racinet's construction of the Double Shuffle Lie algebra in terms of crossed products, connecting it with stabilizers of coproducts and explicit group schemes.

Findings

Identifies the coproduct with a structure on a module over an algebra

Shows the stabilizer of Racinet's coproduct is contained in a larger stabilizer

Provides an explicit group scheme containing the Double Shuffle Lie algebra stabilizer

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 $\hat{\Delta}^{\mathcal{M}}_G$ defined on a module $\hat{\mathcal{M}}_G$ over an algebra $\hat{\mathcal{W}}_G$, which is equipped with its own coproduct $\hat{\Delta}^{\mathcal{W}}_G$, 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 $\hat{\Delta}^{\mathcal{M}}_G$, hence $\mathsf{DMR}_0^G$, is contained in the stabilizer of $\hat{\Delta}^{\mathcal{W}}_G$. This yields an explicit group scheme containing $\mathsf{DMR}_0^G$, which we also express in the Racinet formalism.