On Generalized Cluster Categories

Cluster categories have been introduced by Buan, Marsh, Reineke, Reiten and Todorov in order to categorify Fomin-Zelevinsky cluster algebras. This survey motivates and outlines the construction of a generalization of cluster categories, and explains different applications of these new categories in representation theory.Comment: survey 54pages, v2: small improvements, published in the proceedings of ICRA XIV "Representations of Algebras and Related Topics", European Mathematical Societ

A derived equivalence between cluster equivalent algebras

Let $Q$ be an acyclic quiver. Associated with any element $w$ of the Coxeter group of $Q$, triangulated categories \underline{\Sub}\Lambda_w were introduced in \cite{Bua2}. There are shown to be triangle equivalent to generalized cluster categories \Cc_{\Gamma_w} associated to algebras $\Gamma_w$ of global dimension $\leq 2$ in \cite{ART}. For $w$ satisfying a certain property, called co-$c$-sortable, other algebras $A_w$ of global dimension $\leq 2$ are constructed in \cite{AIRT} with a triangle equivalence \Cc_{A_w}\simeq \underline{\Sub}\Lambda_w. The main result of this paper is to prove that the algebras $\Gamma_w$ and $A_w$ are derived equivalent when $w$ is co-$c$-sortable. The proof uses the 2-APR-tilting theory introduced in \cite{IO}.Comment: to appear in Journal of Algebr

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 $\leq 2$. 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).Comment: 46 pages, small typos as it will appear in Annales de l'Institut Fourie

On the structure of triangulated category with finitely many indecomposables

We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category is of the form $\mathbb{Z}\Delta/G$ where $\Delta$ is a disjoint union of simply laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $\mathbb{Z}\Delta$. Then we prove that for `most' groups $G$, the category \T is standard, \emph{i.e.} $k$-linearly equivalent to an orbit category \mathcal{D}^b(\modd k\Delta)/\Phi. This happens in particular when \T is maximal $d$-Calabi-Yau with $d\geq2$. Moreover, if \T is standard and algebraic, we can even construct a triangle equivalence between \T and the corresponding orbit category. Finally we give a sufficient condition for the category of projectives of a Frobenius category to be triangulated. This allows us to construct non standard 1-Calabi-Yau categories using deformed preprojective algebras of generalized Dynkin type