Publications (12)0 Total impact
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ABSTRACT: In this paper, we give sufficient properties for a finite dimensional graded algebra to be a higher preprojective algebra. These properties are of homological nature, they use Gorensteiness and bimodule isomorphisms in the stable category of CohenMacaulay modules. We prove that these properties are also necessary for 3preprojective algebras using \cite{Kel11} and for preprojective algebras of higher representation finite algebras using \cite{Dugas}.07/2013;  [show abstract] [hide abstract]
ABSTRACT: By Auslander's algebraic McKay correspondence, the stable category of CohenMacaulay modules over a simple singularity is equivalent to the 1cluster category of the path algebra of a Dynkin quiver (i.e. the orbit category of the derived category by the action of the AuslanderReiten translation). In this paper we give a systematic method to construct a similar type of triangle equivalence between the stable category of CohenMacaulay modules over a Gorenstein isolated singularity $R$ and the generalized (higher) cluster category of a finite dimensional algebra $\Lambda$. The key role is played by a bimodule CalabiYau algebra, which is the higher Auslander algebra of $R$ as well as the higher preprojective algebra of an extension of $\Lambda$. As a byproduct, we give a triangle equivalence between the stable category of graded CohenMacaulay $R$modules and the derived category of $\Lambda$. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.04/2011; 
Article: On Generalized Cluster Categories
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ABSTRACT: Cluster categories have been introduced by Buan, Marsh, Reineke, Reiten and Todorov in order to categorify FominZelevinsky 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.01/2011;  [show abstract] [hide abstract]
ABSTRACT: Associated with a finitedimensional algebra of global dimension at most 2, a generalized cluster category was introduced in Amiot (2009) [1]. It was shown to be triangulated, and 2Calabi–Yau when it is Homfinite. By definition, the cluster categories of Buan et al. (2006) [4] are a special case. In this paper we show that a large class of 2Calabi–Yau triangulated categories, including those associated with elements in Coxeter groups from Buan et al. (2009) [7], are triangle equivalent to generalized cluster categories. This was already shown for some special elements in Amiot (2009) [1].Advances in Mathematics. 01/2011;  [show abstract] [hide abstract]
ABSTRACT: Cluster categories of hereditary algebras have been introduced as orbit categories of their derived categories. Keller has pointed out that for nonhereditary algebras orbit categories need not be triangulated, and he introduced the notion of triangulated hull to overcome this problem. In this paper we study the image if the natural functor from the bounded derived category to the cluster category, that is we investigate how far the orbit category is from being the cluster category. We show that for wide classes of nonpiecewise hereditary algebras the orbit category is never equal to the cluster category. Comment: 21 pages10/2010;  [show abstract] [hide abstract]
ABSTRACT: In this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of type $\widetilde{A}$. We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster type $\A_n$ for each possible orientation of $\A_n$. We give an explicit way to read off in which derived equivalence class such an algebra lies, and describe the AuslanderReiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.09/2010;  [show abstract] [hide abstract]
ABSTRACT: In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce the notion of cluster equivalence for these algebras, based on whether their generalized cluster categories are equivalent. We are particularly interested in the question how much information about an algebra is preserved in its generalized cluster category, or, in other words, how closely two algebras are related if they have equivalent generalized cluster categories. Our approach makes use of the clustertilting objects in the generalized cluster categories: We first observe that clustertilting objects in generalized cluster categories are in natural bijection with clustertilting subcategories of derived categories, and then prove a recognition theorem for the latter. Using this recognition theorem we give a precise criterion when two cluster equivalent algebras are derived equivalent. For a given algebra we further describe all the derived equivalent algebras which have the same canonical cluster tilting object in their generalized cluster category. Finally we show that in general, if two algebras are cluster equivalent, then (under certain conditions) the algebras can be graded in such a way that the categories of graded modules are derived equivalent. To this end we introduce mutation of graded quivers with potential, and show that this notion reflects mutation in derived categories.03/2010;  [show abstract] [hide abstract]
ABSTRACT: Let $Q$ be an acyclic quiver and $\Lambda$ be the complete preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama, Reiten and Scott have introduced and studied in \cite{Bua2} a finite dimensional algebra $\Lambda_w=\Lambda/I_w$. In this paper we look at filtrations of $\Lambda_w$ associated to any reduced expression $\mathbf{w}$ of $w$. We are especially interested in the case where the word $\mathbf{w}$ is $c$sortable, where $c$ is a Coxeter element. In this situation, the consecutive quotients of this filtration can be related to tilting $kQ$modules with finite torsionfree class.02/2010;  [show abstract] [hide abstract]
ABSTRACT: In this paper, we study algebras of tame acyclic cluster type. These are algebras of global dimension at most 2 whose generalized cluster category is equivalent to a cluster category of an acyclic quiver of tame representation type. We are particularly interested in their derived equivalence classification. We prove that the algebras which are cluster equivalent to a tree quiver are derived equivalent to the path algebra of this tree. Then we complete the classification by describing explicitly the algebras of cluster type $\A_n$ for each possible orientation of $A_n$. We give an explicit way to read off in which derived equivalence class these algebras are, and describe the AuslanderReiten quiver of their derived categories.01/2010;  [show abstract] [hide abstract]
ABSTRACT: 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 2APRtilting theory introduced in \cite{IO}.11/2009;  [show abstract] [hide abstract]
ABSTRACT: Let $k$ be a field and $A$ a finitedimensional $k$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 2CY and endowed with a canonical clustertilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by GeissLeclercSchr{\"o}er and by BuanIyamaReitenScott. 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 Jacobifinite we prove that it is endowed with a clustertilting 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 Fourier05/2008;  [show abstract] [hide abstract]
ABSTRACT: We study the problem of classifying triangulated categories with finitedimensional 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 AuslanderReiten 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 $d$CalabiYau 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 $1$CalabiYau categories using deformed preprojective algebras of generalized Dynkin type.01/2006;
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2011

Northeastern University
 Department of Mathematics
Boston, MA, United States
