97 research outputs found
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 be an acyclic quiver. Associated with any element of the Coxeter
group of , 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
of global dimension in \cite{ART}. For satisfying a
certain property, called co--sortable, other algebras of global
dimension 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 and are derived equivalent when
is co--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 be a field and a finite-dimensional -algebra of global
dimension . We construct a triangulated category \Cc_A associated to
which, if is hereditary, is triangle equivalent to the cluster category
of . 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 . 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
where is a disjoint union of simply laced Dynkin
diagrams and a weakly admissible group of automorphisms of
. Then we prove that for `most' groups , the category \T
is standard, \emph{i.e.} -linearly equivalent to an orbit category
\mathcal{D}^b(\modd k\Delta)/\Phi. This happens in particular when \T is
maximal -Calabi-Yau with . 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
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