Gentle algebras arising from surface triangulations
Ibrahim Assem Thomas Brüstle Gabrielle Charbonneau-Jodoin Pierre-Guy Plamondon Ibrahim Assem is partially supported by NSERC of Canada and the Université de Sherbrooke. Gabrielle Charbonneau was working under a summer research fellowship of NSERC. Thomas Brüstle is partially supported by NSERC, by Bishop’s University and the Université de Sherbrooke. Pierre-Guy Plamondon was supported by an NSERC graduate fellowship
Abstract
In this paper, we associate an algebra $A(\Gamma)$ to a triangulation $\Gamma$ of a surface $S$ with a set of boundary marking points. This algebra $A(\Gamma)$ is gentle and Gorenstein of dimension one. We also prove that $A(\Gamma)$ is cluster-tilted if and only if it is cluster-tilted of type $\mathbb{A}$ or $\widetilde{\mbox{$\mathbb{A}$}}$ , or if and only if the surface $S$ is a disc or an annulus. Moreover all cluster-tilted algebras of type $\mathbb{A}$ or $\widetilde{\mbox{$\mathbb{A}$}}$ are obtained in this way.
中文速览
带边界标记点的曲面（surface with boundary marked points）的三角剖分（triangulation）可以自然地对应到一类代数结构，但人们并不清楚这类代数究竟覆盖哪些已知的代数类型、以及在什么条件下它们属于重要的丛倾斜代数（cluster-tilted algebra）。作者对每个三角剖分 Γ 构造了一个温和代数（gentle algebra）A(Γ)，并证明它总是一维 Gorenstein 代数；进一步完整刻画了 A(Γ) 成为丛倾斜代数的充要条件——恰好当且仅当底层曲面是圆盘或环形，此时 A(Γ) 恰好给出所有 A 型和 Ã 型丛倾斜代数，而在其他曲面上得到的则是更广泛的温和代数。这一结果将曲面组合几何与代数表示论精确地衔接起来，既为丛倾斜代数提供了清晰的几何分类，也揭示了温和代数作为比丛倾斜代数更大一类结构的本质来源。
原文 arXiv:0903.3347;中英对照 + 大白话阅读 https://aha.fim.ai/paper/0903.3347v2