ON AUTO-EQUIVALENCES AND COMPLETE DERIVED INVARIANTS OF GENTLE ALGEBRAS
Sebastian Opper Mathematisches Institut, Universität Paderborn, Warburger Str. 100, 33098 Paderborn.
Abstract
We study triangulated categories which can be modeled by an oriented marked surface $\mathcal{S}$ and a line field $\eta$ on $\mathcal{S}$ . This includes bounded derived categories of gentle algebras and – conjecturally – all partially wrapped Fukaya categories introduced by Haiden-Katzarkov-Kontsevich [22]. We show that triangle equivalences between such categories induce diffeomorphisms of the associated surfaces preserving orientation, marked points and line fields up to homotopy. This shows that the pair $(\mathcal{S},\eta)$ is a triangle invariant of such categories and prove that it is a complete derived invariant for gentle algebras of arbitrary global dimension. We deduce that the group of auto-equivalences of a gentle algebra is an extension of the stabilizer subgroup of $\eta$ in the mapping class group and a group, which we describe explicitely in case of triangular gentle algebras. We show further that diffeomorphisms associated to spherical twists are Dehn twists.
中文速览
论文研究的是一类能用带标记点的曲面和线场来描述的三角范畴，核心问题是范畴等价或自同构究竟能否被曲面上的几何变换准确反映出来。作者把不可分解对象对应到曲线，并通过弧复形、三角剖分以及对象之间的交互来构造和验证这种几何对应，证明三角等价会诱导保持方向、标记点和线场同伦类的微分同胚。由此得到，曲面连同线场是这类范畴的完整不变量，尤其两个 gentle 代数导出等价，当且仅当它们的曲面模型之间存在保持这些结构的微分同胚；同时，自同构群可分解为曲面映射类群中稳定线场的部分与代数性的部分，球面对象对应的球面扭转则几何上正是沿相应闭曲线的 Dehn 扭转。这个结果把抽象的导出范畴问题转化为可视化的曲面拓扑问题，并为 gentle 代数的分类及其自同构研究提供了统一工具。
原文 arXiv:1904.04859;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1904.04859v1