AI 中文总结
本文证明存在连通的非半单遗传代数是 Ringel 自对偶的，并通过 Auslander-Reiten 理论给出组合刻画，确定其 Dynkin 图类型为 D_{2m} 和 A_{4t+1}。
AI 中文摘要
具有有限多个单模的最高权范畴对应于拟遗传代数。Ringel 对偶性表明，在 Morita 等价意义下，拟遗传代数是成对出现的，而 Ringel 自对偶性则是一个拟遗传代数与其自身配对的现象。本文的主要目的是证明存在连通的非半单遗传代数是 Ringel 自对偶的，并给出 Ringel 自对偶性在 Auslander-Reiten 理论中的组合解释。设 $A$ 是代数闭域 $k$ 上的基本连通遗传代数，并带有拟遗传结构。我们证明，若 $A$ 具有有限表示型且为 Ringel 自对偶，则特征倾斜模 $T$ 的不可分解直和项可由不可分解投射模和不可分解内射模通过应用 Auslander-Reiten 平移的幂次得到，其方式由底层的 Dynkin 图的自同构所支配。特别地，不可分解 $A$-模的个数与单 $A$-模的个数具有相同的奇偶性。反之，我们证明若 $T\cong \tau^{-n}(A)\cong \tau^n\Hom_k(A, k)$ 对某个自然数 $n$ 成立，则 $A$ 是 Ringel 自对偶的且具有有限表示型。利用这些结果，我们确定了允许一个定向和一个拟遗传结构使得对应的路代数是 Ringel 自对偶的单连通 Dynkin 图：它们恰好是 $D_{2m}$（$m\geq 2$）型和 $A_{4t+1}$（$t\geq 0$）型。特别地，没有例外类型出现，也没有连通的非半单遗传 Nakayama 代数是 Ringel 自对偶的。
英文摘要
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.
Comments30 pages, comments are welcome!
