{
  "abstract": "Highest weight categories with finitely many simples correspond to quasi-hereditary algebras. Ringel duality shows that, up to Morita equivalence, quasi-hereditary algebras come in pairs, while Ringel self-duality is the phenomenon in which a quasi-hereditary algebra is paired with itself. The main purpose of this paper is to show that there are connected non-semisimple hereditary algebras that are Ringel self-dual and to give a combinatorial interpretation of Ringel self-duality in terms of Auslander-Reiten theory. Let $A$ be a basic connected hereditary algebra over an algebraically closed field $k$ equipped with a quasi-hereditary structure. We show that, if $A$ is of finite representation type and Ringel self-dual, then the indecomposable summands of the characteristic tilting module $T$ are obtained from the indecomposable projectives, and from indecomposable injectives by applying powers of the Auslander-Reiten translation, in a way governed by an automorphism of the underlying Dynkin diagram. In particular, the number of indecomposable $A$-modules has the same parity as the number of simple $A$-modules. Conversely, we show that if $T\\cong τ^{-n}(A)\\cong τ^n\\Hom_k(A, k)$ for some natural number $n$, then $A$ is Ringel self-dual and of finite representation type. Using these results, we determine the simply laced Dynkin diagrams admitting an orientation and a quasi-hereditary structure for which the corresponding path algebra is Ringel self-dual: they are exactly those of type $D_{2m}$ with $m\\geq 2$ and $A_{4t+1}$ with $t\\geq 0$. In particular, no exceptional type occurs, and no connected non-semisimple hereditary Nakayama algebra is Ringel self-dual.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29942",
  "author_total": 1,
  "authors": [
    {
      "name": "Tiago Cruz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0bvm6817j2108.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29942",
  "fulltext_url": null,
  "id": "0bvm6817j2108",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29942",
  "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.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.29942",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0bvm6817j2108.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29942",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29942"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Ringel self-duality of hereditary algebras via Auslander-Reiten theory",
  "title_text": "Ringel self-duality of hereditary algebras via Auslander-Reiten theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/0bvm6817j2108",
  "venue": "arXiv",
  "versions": []
}