Cluster Categories Coming from Cyclic Posets

Abstract

Cyclic posets are generalizations of cyclically ordered sets. In this article, we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The stable category of a Frobenius category is always triangulated and has a cluster structure in many cases. The continuous cluster categories of [14], the infinity-gon of [12], and the m-cluster category of type A _∞ (m ≥ 3) [13] are examples of this construction as well as some new examples such as the cluster category of ℤ². An extension of this construction and further examples are given in [16].

Key Words:

• Cluster category
• Frobenius category
• Matrix factorization
• Triangulated category

2010 Mathematics Subject Classification:

• 18E30
• 16G20

ACKNOWLEDGMENT

This article was partially written while the second author was at the Mathematical Sciences Research Institute (MSRI) in Berkeley. Adam-Christiaan van Roosmalen gave us some very helpful suggestions at the beginning of this project, and the referee also gave us helpful suggestions at the end.

Notes

Communicated by E. Kirkman