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
代数领域有一类"温和代数"（gentle algebra），人们很早就知道可以用带标记点的曲面来描述它的很多代数信息，但一直没有搞清楚：两个温和代数的有界导出范畴（bounded derived category）等价，是否恰好等价于它们的曲面模型相同？这篇论文引入了"曲面型范畴"（surface-like category）这一框架，证明了任何两个这样的范畴之间的三角等价都会诱导出对应曲面之间保持定向、标记点和线场（line field）同伦类的微分同胚，从而把带线场的标记曲面确立为这类范畴的三角不变量。在此基础上，作者完整证明了任意整体维数的温和代数的导出等价分类定理：两个温和代数导出等价，当且仅当它们的带线场曲面模型微分同胚；同时还精确刻画了温和代数自等价群的结构，并证明球面扭转（spherical twist）对应曲面上的德恩扭转（Dehn twist）。这一结果不仅终结了温和代数导出分类的长期悬案，还将抽象的代数等价问题完全转化为直观的曲面拓扑语言，为同调镜像对称中的部分缠绕深谷范畴（partially wrapped Fukaya category）的研究提供了重要理论基础。
原文 arXiv:1904.04859;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1904.04859v1