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=₍ of a group scheme DMR₀^G attached to a finite abelian group G. Then Enriquez and Furusho proved that DMR₀^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 ^M₆ defined on a module M₆ over an algebra W₆, which is equipped with its own coproduct ^W₆, and the group action on M₆ extends to a compatible action of W₆. We then show that the stabilizer of ^M₆, hence DMR₀^G, is contained in the stabilizer of ^W₆ thus generalizing a result of Enriquez and Furusho Selecta Math. (N. S. ) 29 (2023), article no. 3. This yields an explicit group scheme containing DMR₀^G, which we also express in the Racinet formalism.