Gentle algebras arising from surface triangulations

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

doi:10.2140/ant.2010.4.201

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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

DOI: 10.2140/ant.2010.4.201

Abstract

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

Milestones

Received: 8 April 2009

Revised: 24 June 2009

Accepted: 6 August 2009

Published: 26 January 2010

Authors

Ibrahim Assem
Département de Mathématiques Université de Sherbrooke Sherbrooke, QC J1K 2R1 Canada

Thomas Brüstle
Département de Mathématiques Université de Sherbrooke Sherbrooke, QC J1K 2R1 Canada	Department of Mathematics Bishop’s University College St. Sherbrooke, QC J1M 0C8 Canada

Gabrielle Charbonneau-Jodoin
Département de Mathématiques Université de Sherbrooke Sherbrooke, QC J1K 2R1 Canada

Pierre-Guy Plamondon
Institut de Mathématiques Université Denis Diderot (Paris VII) Case 7012-2, place Jussieu 75251 Paris Cedex 05 France

Vol. 4, No. 2, 2010