{
  "abstract": "We establish a finite system of consistency equations for triangle relation planar algebras. Within the Yang--Baxter relation class, we characterize triangle relations by $H^2(K;\\mathbb C)=0$, where $K$ is the fusion complex recording the support of the fusion coefficients. Together with a reconstruction theorem, this gives an algebraic classification procedure. We classify bi-commutative triangle relation planar algebras with $\\dim P_{2,\\pm}\\leq5$. In ranks two, three, and four, the classification yields respectively $2$, $7$, and $19$ realizable fusion-support pair strata, up to relabelling and shading reversal. These results recover the low-rank examples of Bisch, Jones, Liu, and Lu, and extend the rank-four classification beyond exchange relations. In rank five, we obtain two one-parameter families with different duality types whose generic members do not arise from representations of groups or quantum groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02714",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiaqi Li"
    },
    {
      "name": "Zhengwei Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/arbb1fk86n5tx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02714",
  "fulltext_url": null,
  "id": "arbb1fk86n5tx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02714",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02714",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/arbb1fk86n5tx.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02714",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02714"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Triangle Relation Planar Algebras: Cohomology, Classification, and New Families",
  "title_text": "Triangle Relation Planar Algebras: Cohomology, Classification, and New Families",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/arbb1fk86n5tx",
  "venue": "arXiv",
  "versions": []
}