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

2021/12/28 by Yaddaden, Khalef · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.14140

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=μ_N 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 Δ^M_G defined on a module M_G over an algebra W_G, which is equipped with its own coproduct Δ^W_G, and the group action on M_G extends to a compatible action of W_G. We then show that the stabilizer of Δ^M_G, hence DMR₀^G, is contained in the stabilizer of Δ^W_G. This yields an explicit group scheme containing DMR₀^G, which we also express in the Racinet formalism.

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