A Geometric Interpretation of the Triangulated Structure of m-Cluster Categories

Abstract

The aim of this note is to answer several open problems arising from the geometric description of the m-cluster categories of type A _n and their realization in terms of the mth power of a translation quiver. In particular, we give a geometric interpretation of the triangulated structure of m-cluster categories. Furthermore, we characterize all the connected components arising from a cluster category when taking the mth power of its Auslander–Reiten (AR) quiver.

Key Words:

• Auslander–Reiten quivers
• Auslander–Reiten triangles
• m-cluster categories
• mth power of a translation quiver
• Orbit categories
• Triangulated structure

2000 Mathematics Subject Classification:

• 16G20
• 16G70
• 18E30

ACKNOWLEDGMENTS

I would like to thank Prof. K. Baur for introducing me to such an interesting field, for the inspiring discussions we had and for her time in reading my drafts.

Notes

Communicated by Q. Wu.