{
  "abstract": "We study the completed Jacobi algebras of $x^3+xy^3+p(y)$ for $\\operatorname{ord}_yp=n\\ge4$. The explicit normal-form morsification, the filtered length and centre proofs, and the cyclic terminal identities give finite free semisimple deformations. For every parameter of finite order we construct a polynomial curve neighbourhood and a local projective small contraction; the resulting geometric GV and Donaldson--Thomas calculations identify the block multiplicities. The one-dimensional limit $E_{6,\\infty}$ is studied separately as an order: we determine its normalization, conductor, radical-idealizer chain, and maximal overorders. The canonical chain yields a nine-dimensional finite quotient whose Ringel dual is isomorphic to its opposite. An expanded construction gives a choice-independent $22$-dimensional quotient and a $40$-dimensional Ringel dual; a corner quotient relates the two finite constructions. The length and centre formulas are proved by a filtered comparison with a homogeneous matrix order and by terminal identities at the exceptional endpoint.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13745",
  "author_total": 1,
  "authors": [
    {
      "name": "Shengyong Pan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eef2ascsxx5n6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13745",
  "fulltext_url": null,
  "id": "eef2ascsxx5n6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13745",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7221521023",
  "pdf_url": "https://arxiv.org/pdf/2610.13745",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eef2ascsxx5n6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13745",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13745"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The $E_6$ normal forms: semisimple deformations, geometry, DT invariants, and the $E_{6,\\infty}$ order",
  "title_text": "The E_6 normal forms: semisimple deformations, geometry, DT invariants, and the E_6,∞ order",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/eef2ascsxx5n6",
  "venue": "arXiv",
  "versions": []
}