Crossed Product Interpretation of the Double Shuffle Lie Algebra Attached to a Finite Abelian Group

Racinet studied a 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 \operatorname{\mathsf{DMR}}_{0}^{G} attached to a finite abelian group G . Then Enriquez and Furusho proved that \operatorname{\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 \widehat{\Delta}^{\mathcal{M}}_{G} defined on a module \widehat{\mathcal{M}}_{G} over an algebra \widehat{\mathcal{W}}_{G} , which is equipped with its own coproduct \widehat{\Delta}^{\mathcal{W}}_{G} , and the group action on \widehat{\mathcal{M}}_{G} extends to a compatible action of \widehat{\mathcal{W}}_{G} . We then show that the stabilizer of \widehat{\Delta}^{\mathcal{M}}_{G} , hence \operatorname{\mathsf{DMR}}_{0}^{G} , is contained in the stabilizer of \widehat{\Delta}^{\mathcal{W}}_{G} thus generalizing a result of Enriquez and Furusho [Selecta Math. (N.S.) 29 (2023), article no. 3]. This yields an explicit group scheme containing \operatorname{\mathsf{DMR}}_{0}^{G} , which we also express in the Racinet formalism.