{
  "abstract": "Let $G$ be a complex reductive affine algebraic group with its symmetric tensor category $\\mathcal{C}_G$ of finite-dimensional rational representations. We prove that every simple commutative algebra object with at most countable dimension in the direct limit completion $\\operatorname{Ind}(\\mathcal{C}_G)$ of $\\mathcal{C}_G$ is isomorphic to the algebra $\\mathcal{O}(G/H)$ of regular functions on the homogeneous space $G/H$, for some reductive algebraic subgroup $H$ of $G$. Then we apply this result to the theory of vertex operator algebra extensions. In particular, assuming the strong rationality of the $ \\operatorname{A}_5$-orbifold of the vertex operator algebra $V_{\\mathcal{L}_2}$ associated with the rank-one root lattice $\\mathcal{L}_2:= \\sqrt{2}\\mathbb{Z}$, we classify all the not necessarily rational simple CFT type preunitary vertex operator algebra extensions of the simple unitary Virasoro vertex operator algebra $L(1,0)$ with central charge $c=1$ satisfying a certain spectrum condition. This result is the vertex operator algebra analogue of a conformal net result by Feng Xu. Every strongly rational preunitary vertex operator algebra extension of $L(1,0)$ satisfies the above spectrum condition because of the congruence subgroup modularity property of its characters. As a consequence, we get a complete classification result for strongly rational $c=1$ vertex operator algebras, up to the strong rationality of $V_{\\mathcal{L}_2}^{\\operatorname{A}_5}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00347",
  "author_total": 3,
  "authors": [
    {
      "name": "Sebastiano Carpi"
    },
    {
      "name": "Tiziano Gaudio"
    },
    {
      "name": "Luca Giorgetti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/akqmx1hqqy5je.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00347",
  "fulltext_url": null,
  "id": "akqmx1hqqy5je",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00347",
  "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",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7206152810",
  "pdf_url": "https://arxiv.org/pdf/2609.00347",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/akqmx1hqqy5je.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:00Z",
      "native_id": "2609.00347",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00347"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Algebra objects in direct limit completions of compact Lie group duals and the classification of $c=1$ vertex operator algebras",
  "title_text": "Algebra objects in direct limit completions of compact Lie group duals and the classification of c=1 vertex operator algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/akqmx1hqqy5je",
  "venue": "arXiv",
  "versions": []
}