Cluster categories

Aslak Bakke Buan

For a finite quiver without oriented cycles, there is an associated path algebra (over some field), which is a finite dimensional hereditary algebra. The module category of the path algebra can be extended to a triangulated category called the cluster category. This category is constructed as a certain orbit-category of the bounded derived category of the module category.

The main motivations for studying cluster categories are:

• They are used to model acyclic cluster algebras, where cluster variables are modelled by rigid objects and clusters by maximal rigid (or cluster-tilting) objects.
• The endomorphism rings of cluster-tilting objects are finite dimensional algebras with several interesting properties: They are uniquely determined by their Gabriel quivers. Their representation theory is well understood. They have nice homological properties.

The emphasis will be on the construction of cluster categories and the above mentioned aspects. In addition, a notion of higher cluster categories and a combinatorial interpretation of these will be discussed.

This series will be based on work by many authors.