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

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 $T\cong \tau^{-n}(A)\cong \tau^n\Hom_k(A, k)$ 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.