Muze Ren

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math.QA2026

Double shuffle relations and double antipodes

Muze Ren

In this note, we prove that for every Lie series $ψ$ with no terms of degree less than 3, the relation $[ψ (x, y), x] + [ψ (- x - y, y), - x - y] = 0$ is equivalent to $S_{*} (ψ_{*}) = - ψ_{*}$ , where $ψ_…

math.QA2026

Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra

Muze Ren

Schneps \cite{Schneps2012,Schneps2025} and Enriquez-Furusho \cite{EF4} proved that the double shuffle Lie algebra $dmr_{0}$ embeds into the Kashiwara--Vergne Lie algebra $…