{
  "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 $\\mathfrak{dmr}_0$ satisfies the relation $[ψ(x,y),x]+[ψ(-x-y,y),-x-y]=0$. We further prove that $\\mathfrak{dmr}_0$ injects into the symmetric Kashiwara--Vergne Lie algebra $\\mathfrak{krv}^{\\mathrm{sym}}_2$ of Alekseev and Torossian.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.28163",
  "author_total": 1,
  "authors": [
    {
      "name": "Muze Ren"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/38hye50yaag2g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.28163",
  "fulltext_url": null,
  "id": "38hye50yaag2g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.28163",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7171995607",
  "pdf_url": "https://arxiv.org/pdf/2607.28163",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/38hye50yaag2g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2607.28163",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.28163"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Double shuffle relations and double antipodes",
  "title_text": "Double shuffle relations and double antipodes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/38hye50yaag2g",
  "venue": "arXiv",
  "versions": []
}