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

Let k be a field and A a finite-dimensional k -algebra of global dimension ≤ 2 . We construct 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, we prove that 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. Our results also apply to quivers with potential. Namely, we introduce a cluster category 𝒞 ( Q , W ) associated to a quiver with potential ( Q , W ) . When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra 𝒥 ( Q , W ) .