标题： 倾斜理论方法在准遗传结构中的应用 显示英文标题

标题： Tilting theoretic approach to quasi-hereditary structures

摘要： 一个拟半单代数是一个在其单模上配备某种偏序 $\unlhd$的代数。 这种偏序——称为拟半单结构——通过 Ringel 的一个经典结果产生一个特征倾斜模 $T_{\unlhd}$。 一个基本问题是确定哪些倾斜模可以被实现为特征倾斜模。 我们通过使用 IS-倾斜模的概念来回答这个问题，IS-倾斜模是一个对 $(T,\unlhd)$，其中包括一个倾斜模 $T$和对其直和项上的一个偏序 $\unlhd$，使得沿着 $\unlhd$进行迭代幂等截断时总是会揭示出一个单的直和项。 具体来说，我们证明了当且仅当存在某个$\unlhd$使得$(T,\unlhd)$是 IS-tilting 模时，倾斜模$T$是特征的；在这种情况下，我们有$T=T_{\unlhd}$。这个结果使我们能够使用倾斜理论来研究拟遗传结构。作为上述结果的应用，我们证明了对于代数$A$，所有倾斜模都是特征的当且仅当$A$是一个二次线性 Nakayama 代数。此外，对于这样的$A$，我们提供了其倾斜模集合的一个分解，该分解可用于推导枚举其拟遗传结构的递归公式。 最后，我们通过“节点粘合”和二叉树序列描述$A$的拟半单结构。 显示英文摘要

摘要： A quasi-hereditary algebra is an algebra equipped with a certain partial order $\unlhd$ on its simple modules. Such a partial order -- called a quasi-hereditary structure -- gives rise to a characteristic tilting module $T_{\unlhd}$ by a classical result due to Ringel. A fundamental question is to determine which tilting modules can be realised as characteristic tilting modules. We answer this question by using the notion of IS-tilting module, which is a pair $(T,\unlhd)$ of a tilting module $T$ and a partial order $\unlhd$ on its direct summands such that iterative idempotent truncation along $\unlhd$ always reveals a simple direct summand. Specifically, we show that a tilting module $T$ is characteristic if, and only if, there is some $\unlhd$ so that $(T,\unlhd)$ is IS-tilting; in which case, we have $T=T_{\unlhd}$. This result enables us to study quasi-hereditary structures using tilting theory. As an application of the above result, we show that, for an algebra $A$, all tilting modules are characteristic if, and only if, $A$ is a quadratic linear Nakayama algebra. Furthermore, for such an $A$, we provide a decomposition of the set of its tilting modules that can be used to derive a recursive formula for enumerating its quasi-hereditary structures. Finally, we describe the quasi-hereditary structures of $A$ via `nodal gluing' and binary tree sequences.