gentle 代数的倾斜突变及其组合描述
Tilting mutation of gentle algebras and its combinatorial description
- Southwest Jiaotong University（西南交通大学）
- Sichuan University（四川大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过广义 BB-倾斜模研究 gentle 代数的倾斜突变，给出基于箭向和关系的判据，并构造突变后自同态代数的组合描述，证明所得代数仍为 gentle 且导出等价。
AI中文摘要：
倾斜突变通过替换倾斜对象的一个不可分解直和项，提供了一种构造导出等价代数的自然方法。本文通过广义 $\mathrm{BB}$-倾斜模研究 gentle 代数的倾斜突变。对于 gentle 代数 $A=\mathbb{k} Q/\langle I\rangle$ 和顶点 $k\in Q_0$，我们首先给出一个用与 $k$ 关联的箭向和关系表达的充分必要条件，使得 $P(k)$ 对应的最小左逼近产生倾斜突变。该判据统一适用于有环或无环的顶点。当突变存在时，我们确定突变倾斜模的不可分解直和项之间的不可约态射，并利用它们显式构造其自同态代数的 Gabriel 箭图和定义关系。特别地，我们得到 gentle 对的组合突变 $(Q,I)\mapsto(Q',I')$，使得 $\mu_k^+(A)\cong \mathbb{k} Q'/\langle I'\rangle$，从而所得代数仍是 gentle 的且与 $A$ 导出等价。最后，我们通过相反的 gentle 对阐述对偶余倾斜突变及其组合描述。
英文摘要：
Tilting mutation provides a natural way to construct derived-equivalent algebras by replacing an indecomposable summand of a tilting object. Holm, Schröer and Zimmermann~\cite{HSZ01} (see also the appendix of~\cite{AA08}) have described combinatorial transformations of the quiver and relations of a gentle algebra that yield derived equivalences via tilting complexes. In this paper, we develop a self-contained module-theoretic framework for tilting mutation via generalized $\mathrm{BB}$-tilting modules. For a gentle algebra $A=\mathbb{k} Q/\langle I\rangle$ and a vertex $k\in Q_0$, we first give a necessary and sufficient condition, expressed in terms of the arrows and relations incident with $k$, for the corresponding minimal left approximation of $P(k)$ to yield a tilting mutation. This criterion applies uniformly to vertices with or without loops. When the mutation exists, we determine the irreducible morphisms between the indecomposable summands of the mutated tilting module and use them to construct explicitly the Gabriel quiver and defining relations of its endomorphism algebra. In particular, we also provide a combinatorial mutation $(Q,I)\mapsto(Q',I')$ of gentle pairs such that $μ_k^+(A)\cong \mathbb{k} Q'/\langle I'\rangle,$ so the resulting algebra is again gentle and derived equivalent to $A$. Finally, we formulate the dual cotilting mutation and its combinatorial description via opposite gentle pairs.
