Ringel self-duality of hereditary algebras via Auslander-Reiten theory

@misc{cruz2026ringel,
  title = {{Ringel self-duality of hereditary algebras via Auslander-Reiten theory}},
  author = {Tiago Cruz},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29942},
  eprint = {2609.29942},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29942},
}

TY  - UNPB
TI  - Ringel self-duality of hereditary algebras via Auslander-Reiten theory
AU  - Tiago Cruz
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29942
UR  - https://arxiv.org/abs/2609.29942
AB  - 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≅ τ^−n(A)≅ τ^nHom_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≥ 2 and A_4t+1 with t≥ 0. In particular, no exceptional type occurs, and no connected non-semisimple hereditary Nakayama algebra is Ringel self-dual.
ER  -