{
  "abstract": "Let \\(\\mathfrak{g}\\) be any finite-dimensional complex simple Lie algebra, let $h^\\vee$ be its dual Coxeter number, and let \\(e\\) be its highest-root vector which is a minimal nilpotent element. In this paper, using a result of Bavula on invariants of commuting locally nilpotent derivations, we show that \\[ U(\\mathfrak{g})_f \\cong \\mathcal{D}^+_{h^\\vee-2}\\otimes \\mathcal{D}_1\\otimes U(\\mathfrak{g},e), \\] where the first factor is a polynomial Weyl algebra of rank $h^\\vee-2$, the second factor is a Laurent Weyl algebra of rank one, and $U(\\mathfrak{g},e)$ is the finite $W$-algebra associated with $\\mathfrak{g}$ and $e$. As an application, we prove that if $M$ is a simple infinite-dimensional $\\mathfrak{g}$-module with the minimal GK dimension on which $f-1$ acts locally nilpotently, then $M\\cong P\\otimes V$ for some simple $\\mathcal{D}^+_{h^\\vee-2}\\otimes \\mathcal{D}_1$-module $P$ and some finite-dimensional simple $U(\\mathfrak{g},e)$-module $V$. We also give a new proof of the Skryabine quivalence and Petukhov's birational decomposition of the quotient division ring of $U(\\mathfrak{g})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05855",
  "author_total": 1,
  "authors": [
    {
      "name": "Genqiang Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9g1nd1pmtcpxp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05855",
  "fulltext_url": null,
  "id": "9g1nd1pmtcpxp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05855",
  "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.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05855",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9g1nd1pmtcpxp.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05855",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05855"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Root vector localization and minimal nilpotent finite $W$-algebras",
  "title_text": "Root vector localization and minimal nilpotent finite W-algebras",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/9g1nd1pmtcpxp",
  "venue": "arXiv",
  "versions": []
}