On the cluster category of a marked surface without punctures

Brüstle, Thomas; Zhang, Jie

doi:10.2140/ant.2011.5.529

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On the cluster category of a marked surface without punctures

Thomas Brüstle and Jie Zhang

Vol. 5 (2011), No. 4, 529–566

DOI: 10.2140/ant.2011.5.529

Abstract

We study the cluster category $C_{(S, M)}$ of a marked surface $(S, M)$ without punctures. We explicitly describe the objects in $C_{(S, M)}$ as direct sums of homotopy classes of curves in $(S, M)$ and one-parameter families related to noncontractible closed curves in $(S, M)$ . Moreover, we describe the Auslander–Reiten structure of the category $C_{(S, M)}$ in geometric terms and show that the objects without self-extensions in $C_{(S, M)}$ correspond to curves in $(S, M)$ without self-intersections. As a consequence, we establish that every rigid indecomposable object is reachable from an initial triangulation.

Keywords

cluster category, marked surface

Mathematical Subject Classification 2000

Primary: 16G99

Secondary: 16G20, 57N05, 57M50, 16G70

Milestones

Received: 14 May 2010

Revised: 13 August 2010

Accepted: 12 September 2010

Published: 21 December 2011

Proposed: Andrei Zelevinsky

Seconded: Bernd Sturmfels, Victor Reiner

Authors

Thomas Brüstle
Département de Mathématiques Université de Sherbrooke Sherbrooke J1K 2R1 Canada	Department of Mathematics Bishop’s University Sherbrooke J1M 1Z7 Canada

Jie Zhang
Département de Mathématiques Université de Sherbrooke Sherbrooke J1K 2R1 Canada

Vol. 5, No. 4, 2011