On the compatibility of the Betti harmonic coproduct with cyclotomic filtrations

2025/08/17 by Enriquez, Benjamin, Yaddaden, Khalef
#16T10. Secondary 16T05 #16W70 #Algebraic Topology (math.AT) #FOS: Mathematics #Number Theory (math.NT) #Primary 11M32 #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2508.12208

Abstract

In a previous paper, the second author introduced a Betti counterpart of N-cyclotomic double shuffle theory for any N ≥ 1. The construction is based on the group algebra of the free group F₂, endowed with a filtration relative to a morphism F₂ → μ_N (where μ_N is the group of N-th roots of unity). One of the main results therein is the construction of a complete Hopf algebra coproduct \widehatΔ^{W, B}_N on the relative completion of a specific subalgebra W^B of the group algebra of F₂. However, an explicit formula for this coproduct is missing. In this paper, we show that the discrete Betti harmonic coproduct Δ^{W, B} defined in \citeEF1 for the classical case (N=1) by the first author and Furusho remains compatible with the filtration structure on W^B induced by the relative completion for arbitrary N. This compatibility suggests that the completion corresponding to Δ^{W, B} is a candidate for an explicit realization of \widehatΔ^{W, B}_N.

Citations

• The cyclotomic double shuffle torsor in terms of Betti and de Rham coproducts
• Crossed product interpretation of the Double Shuffle Lie algebra attached to a finite Abelian group
• The Betti side of the double shuffle theory. II. Double shuffle relations for associators
• Galois symmetries of fundamental groupoids and noncommutative geometry

Related

• The Betti side of the double shuffle theory. I. The harmonic coproducts
• The cyclotomic double shuffle torsor in terms of Betti and de Rham coproducts
• A categorical formulation of the Deligne-Terasoma approach to double shuffle theory
• A stabilizer interpretation of double shuffle Lie algebras
• The Betti side of the double shuffle theory. II. Double shuffle relations for associators