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