{
  "abstract": "We introduce a quantum deformation $\\mathbf{U}(\\mathfrak{gl}_n[t])$ of the universal enveloping algebra of the current algebra $\\mathfrak{gl}_n[t]$, realized as a Hopf parabolic subalgebra of quantum affine $\\mathfrak{gl}_n$. Unlike the Yangian---the standard quantization of the current algebra---our algebra admits a canonical basis. We give a BLM-type realization of \\(\\mathbf{U}(\\mathfrak{gl}_n[t])\\) via certain subalgebras of affine quantum Schur algebras, and then construct canonical bases for the modified quantum current algebra $\\dot{\\mathbf{U}}(\\mathfrak{gl}_n[t])$ and for its finite dimensional irreducible graded modules. Moreover, we prove a rigidity theorem: every finite dimensional polynomial irreducible module for quantum affine $\\mathfrak{gl}_n$ remains irreducible when restricted to ${\\mathbf U}_{\\mathbf v}(\\frak{gl}_n[t])$ (the specialization of ${\\mathbf U}(\\frak{gl}_n[t])$ at a non-root-of-unity complex number $\\mathbf{v}$); conversely, every finite dimensional affine polynomial irreducible ${\\mathbf U}_{\\mathbf v}(\\frak{gl}_n[t])$-module extends uniquely to a polynomial irreducible module for quantum affine $\\mathfrak{gl}_n$. More generally, we classify all finite dimensional polynomial irreducible \\({\\mathbf U}_{\\mathbf v}(\\frak{gl}_n[t])\\)-modules by \\(\\mathcal Q(n)\\timesΛ^+(n)\\). As an application, we establish a bijection between the isomorphism classes of finite dimensional polynomial irreducible modules of \\({\\bf U}_{\\mathbf v}(\\frak{gl}_n[t])\\) and those of the Yangian \\(Y(\\mathfrak{gl}_n)\\). Thus \\({\\bf U}(\\mathfrak{gl}_n[t])\\) admits canonical bases for its modified form and irreducible graded modules, and its finite dimensional polynomial irreducible modules are in correspondence with those of the Yangian.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.08937",
  "author_total": 1,
  "authors": [
    {
      "name": "Qiang Fu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1rc918hm5dbsc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.08937",
  "fulltext_url": null,
  "id": "1rc918hm5dbsc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.08937",
  "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",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7164019711",
  "pdf_url": "https://arxiv.org/pdf/2606.08937",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1rc918hm5dbsc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2606.08937",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.08937"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Quantum current algebra ${\\bf U}(\\frak{gl}_n[t])$: canonical bases and polynomial representations",
  "title_text": "Quantum current algebra bf U(frakgl_n[t]): canonical bases and polynomial representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/1rc918hm5dbsc",
  "venue": "arXiv",
  "versions": []
}