Cluster categories for algebras of global dimension 2 and quivers with potential

Claire Amiot

Let k be a field and A a finite-dimensional k-algebra of global dimension ≤2. In this talk, I will explain the construction of a triangulated category C_A associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When C_A is Hom-finite, it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Using these results, it is possible to introduce a cluster category C_(Q,W) associated to a quiver with potential (Q,W). When it is Jacobi-finite, it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra J(Q,W).