{
  "abstract": "Let $W_{\\geq -1} = \\mathbb{C}[t]\\partial$ and $W = \\mathbb{C}[t, t^{-1}]\\partial$ be the Witt algebra of algebraic vector fields on $\\mathbb{C}$ and $\\mathbb{C}^*$ respectively. In this paper, we make significant progress toward the open conjecture that the enveloping algebras $\\mathrm{U}(W_{\\geq -1})$ and $\\mathrm{U}(W)$ satisfy the ascending chain condition (ACC) on two-sided ideals. We show that all homomorphic images of $\\mathrm{U}(W_{\\geq -1})$ and $\\mathrm{U}(W)$ under the family of ``orbit homomorphisms'' of arbitrary Gelfand-Kirillov dimension satisfy ACC on ideals. These orbit homomorphisms were the key ingredient allowing us to ``lift'' the Dixmier map from finite-dimensional solvable settings to infinite-dimensional settings of the Witt and Virasoro algebras in our recent work [Pham, 2025, arXiv:2504.14670]. As a result, we completely classify the prime and primitive spectra of these homomorphic images. As these images approximate $\\mathrm{U}(W_{\\geq -1})$ better as their GK-dimension increases, this classification sheds new light on the two-sided and prime ideal structures of $\\mathrm{U}(W_{\\geq -1})$. Finally, we discuss several applications of our results to the Dixmier map for $W_{\\geq -1}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01913",
  "author_total": 1,
  "authors": [
    {
      "name": "Tuan Anh Pham"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4qq6hsqgdtef2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01913",
  "fulltext_url": null,
  "id": "4qq6hsqgdtef2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01913",
  "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.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7216248548",
  "pdf_url": "https://arxiv.org/pdf/2610.01913",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4qq6hsqgdtef2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01913",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01913"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Ideals of homomorphic images of the enveloping algebra of the Witt algebra",
  "title_text": "Ideals of homomorphic images of the enveloping algebra of the Witt algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/4qq6hsqgdtef2",
  "venue": "arXiv",
  "versions": []
}