Stability conditions for cyclic quivers

ABSTRACT

Let Δ_n be a cyclic quiver of n vertices and let 𝒯_n denote the category of nilpotent representations of Δ_n over a field. In the present paper, we study the stability conditions on 𝒯_n and describe the semistable subcategories of 𝒯_n with a fixed slope. This provides an alternative characterization of the thick subcategories of 𝒯_n. Further, we obtain a relationship between the Harder–Narasimhan system for Δ_n and the Ringel–Hall algebra of Δ_n.

KEYWORDS:

• Cyclic quiver
• semistable subcategory
• stability condition

2010 Mathematics Subject Classification:

• 16G20
• 16G70

Acknowledgments

The author thanks Professor Bangming Deng for his encouragement and careful guidance. We also thank Dr. Shiquan Ruan for many helpful discussions. We are very grateful to the referee for helpful comments and valuable suggestions, especially for pointing out some mistakes in an earlier version of the paper.