{
  "abstract": "Let $A$ be a representation-infinite finite-dimensional algebra over an algebraically closed field $k$. We prove that for every infinite cardinal $κ$ there is an indecomposable $A$-module $M$ with $\\dim_k M=κ$. Over algebraically closed fields, this settles Simson's conjecture, also called the third Brauer--Thrall conjecture, that every representation-infinite finite-dimensional algebra has indecomposable modules of arbitrarily large cardinality. To construct these modules, we extend Bongartz's representation embedding from finite-dimensional Kronecker modules without a simple injective summand to such modules of arbitrary dimension. The extended functor commutes with direct sums and filtered colimits and preserves the dimension of infinite-dimensional modules. Applying it to the indecomposable Kronecker modules of every infinite dimension constructed by Göbel and Simson yields the required $A$-modules.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06325",
  "author_total": 1,
  "authors": [
    {
      "name": "Haruhisa Enomoto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/062d1ax9bfsy9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06325",
  "fulltext_url": null,
  "id": "062d1ax9bfsy9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06325",
  "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.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06325",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/062d1ax9bfsy9.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06325",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06325"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Representation embeddings and the third Brauer--Thrall conjecture",
  "title_text": "Representation embeddings and the third Brauer--Thrall conjecture",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/062d1ax9bfsy9",
  "venue": "arXiv",
  "versions": []
}