Vol. 4, No. 2, 2010

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Gentle algebras arising from surface triangulations

Ibrahim Assem, Thomas Brüstle, Gabrielle Charbonneau-Jodoin and Pierre-Guy Plamondon

Vol. 4 (2010), No. 2, 201–229
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 $\stackrel{̃}{\mathbb{A}}$, or if and only if the surface $S$ is a disc or an annulus. Moreover all cluster-tilted algebras of type $\mathbb{A}$ or $\stackrel{̃}{\mathbb{A}}$ are obtained in this way.

Keywords
bordered surface with marked points, triangulated surface, quiver with potential, gentle algebra
Mathematical Subject Classification 2000
Primary: 16S99
Secondary: 16G20, 57N05, 57M50