Double shuffle relations and double antipodes

[2607.28163] Double shuffle relations and double antipodes Skip to main content Search Submit Donate Log in Mathematics > Quantum Algebra arXiv:2607.28163 (math) [Submitted on 30 Jul 2026 ( v1 ), last revised 7 Sep 2026 (this version, v3)] Title: Double shuffle relations and double antipodes Authors: Muze Ren View PDF HTML (experimental) Abstract: In this note, we prove that for every Lie series $\psi$ with no terms of degree less than 3, the relation $$[\psi(x,y),x]+[\psi(-x-y,y),-x-y]=0$$ is equivalent to $S_*(\psi_*)=-\psi_*$, where $\psi_*$ denotes the regularization of $\psi$ 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 $\psi$ in Racinet's double shuffle Lie algebra $\mathfrak{dmr}_0$ satisfies the relation $[\psi(x,y),x]+[\psi(-x-y,y),-x-y]=0$. We further prove that $\mathfrak{dmr}_0$ injects into the symmetric Kashiwara--Vergne Lie algebra $\mathfrak{krv}^{\mathrm{sym}}_2$ of Alekseev and Torossian. Comments: 12 pages, corrected some typos and improved the language, submitted version Subjects: Quantum Algebra (math.QA) Cite as: arXiv:2607.28163 [math.QA] (or arXiv:2607.28163v3 [math.QA] for this version) https://doi.org/10.48550/arXiv.2607.28163 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Muze Ren [ view email ] [v1] Thu, 30 Jul 2026 13:03:56 UTC (24 KB) [v2] Fri, 21 Aug 2026 21:12:06 UTC (28 KB) [v3] Mon, 7 Sep 2026 13:39:27 UTC (29 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.QA < prev | next > new | recent | 2026-07 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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