{
  "abstract": "Let $k$ be an algebraically closed field. The generalized or $n$-Kronecker quiver $K(n)$ is the quiver with two vertices, called a source and a sink, and $n$ arrows from source to sink. We use $\\modd\\cK_n$ to denote the category of the finite-dimensional modules of the path algebra $\\cK_n=kK(n)$. There exist a function called dimension vector $\\dimu: \\modd \\cK_n\\rightarrow \\mathbb{Z}^2; M \\mapsto (\\dim_k M_1, \\dim_k M_2 )$, and two functors $σ: \\mathbb{Z}^2 \\rightarrow \\mathbb{Z}^2; (x,y)\\mapsto (nx-y,x)$, and $δ: \\mathbb{Z}^2\\rightarrow \\mathbb{Z}^2;(x,y)\\mapsto (y,x)$. Then the set $\\mathbf{F}=\\{(x,y)\\mid \\frac{2}{n}x\\leq y\\leq x\\}$ is the fundamental domain of the dimension vectors of regular modules under the action of $δ$ and $σ$ in $\\modd\\cK_n$. We call a regular module $M\\in \\modd\\cK_n$ \\textit{elementary} if there does not exist a short exact sequence $(0)\\rightarrow L\\rightarrow M\\rightarrow N\\rightarrow (0)$ with $L,N$ being non-zero regular. In this note, we focus on the set $\\mathbf{F}$, and when $x<n$, we establish a one-to-one correspondence between elementary modules and spaces of fixed rank.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14238",
  "author_total": 1,
  "authors": [
    {
      "name": "Jie Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d8pbvt8rx7nth.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14238",
  "fulltext_url": null,
  "id": "d8pbvt8rx7nth",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14238",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7213322278",
  "pdf_url": "https://arxiv.org/pdf/2609.14238",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d8pbvt8rx7nth.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14238",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14238"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Elementary modules of the generalized Kronecker quivers K(n)",
  "title_text": "Elementary modules of the generalized Kronecker quivers K(n)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/d8pbvt8rx7nth",
  "venue": "arXiv",
  "versions": []
}