Gentle algebras arising from surface triangulations

Abstract

We associate an algebra $A\left(\Gamma \right)$ to a triangulation $\Gamma$ of a surface $S$ with a set of boundary marking points. This algebra $A\left(\Gamma \right)$ is gentle and Gorenstein of dimension one. We also prove that $A\left(\Gamma \right)$ is cluster-tilted if and only if it is cluster-tilted of type $\mathbb{A}$ or $\tilde{\mathbb{A}}$, or if and only if the surface $S$ is a disc or an annulus. Moreover all cluster-tilted algebras of type $\mathbb{A}$ or $\tilde{\mathbb{A}}$ are obtained in this way.