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Double shuffle relations and double antipodes
Abstract
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* denotes the regularization of ψ and S* is the harmonic antipode. The proof relies on the calculations of the harmonic antipode S* and the shuffle antipode S. As a consequence, we prove that every ψ in Racinet's double shuffle Lie algebra dmr0 satisfies the relation [ψ(x,y),x]+[ψ(−x−y,y),−x−y]=0. We further prove that dmr0 injects into the symmetric Kashiwara--Vergne Lie algebra krvsym2 of Alekseev and Torossian.
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BibTeX
@misc{ren2026double,
title = {{Double shuffle relations and double antipodes}},
author = {Muze Ren},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.28163},
eprint = {2607.28163},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.28163},
}
RIS
TY - UNPB
TI - Double shuffle relations and double antipodes
AU - Muze Ren
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2607.28163
UR - https://arxiv.org/abs/2607.28163
AB - 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_* denotes the regularization of ψ and S_* is the harmonic antipode. The proof relies on the calculations of the harmonic antipode S_* and the shuffle antipode S. As a consequence, we prove that every ψ in Racinet's double shuffle Lie algebra dmr_0 satisfies the relation [ψ(x,y),x]+[ψ(−x−y,y),−x−y]=0. We further prove that dmr_0 injects into the symmetric Kashiwara--Vergne Lie algebra krv^sym_2 of Alekseev and Torossian.
ER -