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

TL;DR

This paper constructs and analyzes new triangulated categories associated with finite-dimensional algebras of global dimension 2 and quivers with potential, generalizing cluster categories and establishing their properties.

Contribution

It introduces a new class of triangulated categories for algebras of global dimension 2 and quivers with potential, extending known cluster categories and identifying conditions for their Calabi-Yau and cluster-tilting structures.

Findings

The categories are 2-Calabi-Yau when Hom-finite.

Existence of canonical cluster-tilting objects in these categories.

Application to Jacobian algebras from quivers with potential.

Abstract

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. Our results also apply to quivers with potential. Namely, we introduce a cluster category $\Cc_{(Q,W)}$ 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 $\Jj(Q,W)$.