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

Understand

Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\leq 2$.

• We construct a triangulated category $\Cc_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$.
• When $\Cc_A$ 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-Schr{\"o}er and by Buan-Iyama-Reiten-Scott.