The cluster complex of an hereditary artin algebra

TL;DR

This paper demonstrates that the cluster complex of any hereditary artin algebra forms an abstract simplicial polytope, revealing a structured geometric interpretation and showing that all cluster-tilting objects are interconnected through mutation.

Contribution

It establishes the polytope structure of the cluster complex for hereditary artin algebras and proves the connectedness of cluster-tilting objects under mutation.

Findings

Cluster complex forms an abstract simplicial polytope.

All cluster-tilting objects are mutation-equivalent.

Provides geometric insight into the structure of hereditary artin algebras.

Abstract

We show that the cluster complex of an arbitrary hereditary artin algebra has the structure of an abstract simplicial polytope. In particular, the cluster-tilting objects form one equivalence class under mutation.