A generalization of cluster categories

Cluster categories were introduced by Buan, Marsh, Reineke, Reiten, and Todorov in order to understand combinatorial aspects of cluster algebras. The "categorification process" gives an important link between cluster algebras and quiver representation. For instance, each cluster algebra $A_Q$, associated to an acyclic quiver $Q$, is "categorified" by a triangulated category $C_Q$: the cluster category. In this talk, I will define all basic notions of cluster algebras, cluster categories and illustrate the "categorification process" by examples. I will then explain how it is possible to generalize the notion of cluster category by replacing the path algebra of the quiver $Q$ by an algebra of global dimension 2.