From m-clusters to m-noncrossing partitions via exceptional sequences (2010)
Buan, Aslak Bakke, Reiten, Idun, Thomas, Hugh
Let W be a finite crystallographic reflection group. The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W, and with the number of...
Three kinds of mutation (2010)
Buan, Aslak Bakke, Reiten, Idun, Thomas, Hugh
For a finite dimensional hereditary algebra, we consider: exceptional sequences in the category of finite dimensional modules, silting objects in the bounded derived category, and m-cluster tilting...
Preprojective algebras and c-sortable words (2010)
Amiot, Claire, Iyama, Osamu, Reiten, Idun, Todorov, Gordana
Let $Q$ be an acyclic quiver and $\Lambda$ be the completion of the preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama,...
Preprojective algebras and c-sortable words (2010)
Amiot, Claire, Iyama, Osamu, Reiten, Idun, Todorov, Gordana
Let $Q$ be an acyclic quiver and $\Lambda$ be the completion of the preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama,...
Preprojective algebras and c-sortable words (2010)
Amiot, Claire, Iyama, Osamu, Reiten, Idun, Todorov, Gordana
Let $Q$ be an acyclic quiver and $\Lambda$ be the completion of the preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama,...
2-Auslander algebras associated with reduced words in Coxeter groups (2010)
In this paper we investigate the endomorphism algebras of standard cluster tilting objects in the stably 2-Calabi-Yau categories $\Sub{\Lambda_w}$ with elements $w$ in Coxeter groups in \cite{BIRSc}....
Preprojective algebras and c-sortable words (2010)
Amiot, Claire, Iyama, Osamu, Reiten, Idun, Todorov, Gordana
Let $Q$ be an acyclic quiver and $\Lambda$ be the completion of the preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama,...
Preprojective algebras and c-sortable words (2010)
Amiot, Claire, Iyama, Osamu, Reiten, Idun, Todorov, Gordana
Let $Q$ be an acyclic quiver and $\Lambda$ be the completion of the preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama,...
Preprojective algebras and c-sortable words (2010)
Amiot, Claire, Iyama, Osamu, Reiten, Idun, Todorov, Gordana
Let $Q$ be an acyclic quiver and $\Lambda$ be the completion of the preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama,...
The ubiquity of generalized cluster categories (2009)
Amiot, Claire, Reiten, Idun, Todorov, Gordana
Associated with some finite dimensional algebras of global dimension at most 2, a generalized cluster category was introduced in \cite{Ami3}, which was shown to be triangulated and 2-Calabi-Yau when...
The ubiquity of generalized cluster categories (2009)
Amiot, Claire, Reiten, Idun, Todorov, Gordana
Associated with some finite dimensional algebras of global dimension at most 2, a generalized cluster category was introduced in \cite{Ami3}, which was shown to be triangulated and 2-Calabi-Yau when...
The ubiquity of generalized cluster categories (2009)
Amiot, Claire, Reiten, Idun, Todorov, Gordana
Associated with some finite dimensional algebras of global dimension at most 2, a generalized cluster category was introduced in \cite{Ami3}, which was shown to be triangulated and 2-Calabi-Yau when...
Cluster-tilted algebras are Gorenstein and stably (2009)
Abstract. We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show...
Cluster tilting for one-dimensional hypersurface singularities (2009)
Igor Burban, Osamu Iyama, Bernhard Keller, Idun Reiten
Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of...
reprint of the 1995 original. [353] (2009)
Ad S. Abeasis, A. Del Fra, Adk S. Abeasis, A. Del Fra, H. Kraft, Ab Valery Alexeev, ...
Numbers in square brackets at the end of each entry indicate the pages in the
Denominators of cluster variables (2009)
Buan, Aslak Bakke, Marsh, Robert J., Reiten, Idun
Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in...
The ubiquity of generalized cluster categories (2009)
Amiot, Claire, Reiten, Idun, Todorov, Gordana
Associated with some finite dimensional algebras of global dimension at most 2, a generalized cluster category was introduced in \cite{Ami3}, which was shown to be triangulated and 2-Calabi-Yau when...
The ubiquity of generalized cluster categories (2009)
Amiot, Claire, Reiten, Idun, Todorov, Gordana
Associated with some finite dimensional algebras of global dimension at most 2, a generalized cluster category was introduced in \cite{Ami3}, which was shown to be triangulated and 2-Calabi-Yau when...
The ubiquity of generalized cluster categories (2009)
Amiot, Claire, Reiten, Idun, Todorov, Gordana
Associated with some finite dimensional algebras of global dimension at most 2, a generalized cluster category was introduced in \cite{Ami3}, which was shown to be triangulated and 2-Calabi-Yau when...
Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras (2008)
American Journal of Mathematics - Volume 130, Number 4, August 2008
Mutation of cluster-tilting objects and potentials (2008)
Buan, Aslak Bakke, Iyama, Osamu, Reiten, Idun, Smith, David
We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between...
Exact Categories and Vector Space Categories (2007)
Peter Dräxler, Peter Dr Axler, Idun Reiten, Øyvind Solberg, Sverre O. Smalø, ��yvind Solberg
. In a series of papers starting with [ASo] additive subbifunctors F of the bifunctor Ext ( ; ) are studied in order to establish a relative homology theory for an artin algebra . On the other hand,...
Denominators of cluster variables (2007)
Buan, Aslak Bakke, Marsh, Robert J., Reiten, Idun
Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in...
Cluster tilting for one-dimensional hypersurface singularities (2007)
Burban, Igor, Iyama, Osamu, Keller, Bernhard, Reiten, Idun
In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster...
Bounded derived categories and repetitive algebras (2007)
Happel, Dieter, Keller, Bernhard, Reiten, Idun
By a theorem due to the first author, the bounded derived category of a finite-dimensional algebra over a field embeds fully faithfully into the stable category over its repetitive algebra. This...
Cluster structures for 2-Calabi-Yau categories and unipotent groups (2007)
Buan, Aslak Bakke, Iyama, Osamu, Reiten, Idun, Scott, Jeanne
We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to...
Acyclic Calabi-Yau categories (2006)
Keller, Bernhard, Reiten, Idun
We show that an algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category if it contains a cluster tilting subcategory whose quiver has no oriented cycles....
Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras (2006)
We say that an algebra $\Lambda$ over a commutative noetherian ring $R$ is Calabi-Yau of dimension $d$ ($d$-CY) if the shift functor $[d]$ gives a Serre functor on the bounded derived category of the...
Acyclic quivers of finite mutation type (2006)
Buan, Aslak Bakke, Reiten, Idun
The theory of cluster algebras gives rise to an important notion of quiver mutations. Using the representation theory of finite-dimensional algebras, we show that the acyclic connected quivers of...
Acyclic quivers of finite mutation type (2006)
Buan, Aslak Bakke, Reiten, Idun
The theory of cluster algebras gives rise to an important notion of quiver mutations. Using the representation theory of finite-dimensional algebras, we show that the acyclic connected quivers of...
Cluster-tilted algebras are Gorenstein and stably Calabi-Yau (2005)
Keller, Bernhard, Reiten, Idun
We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the...
Tame concealed algebras and cluster quivers of minimal infinite type (2005)
Buan, Aslak Bakke, Reiten, Idun, Seven, Ahmet I.
In this paper we explain how and why the list of Happel-Vossieck of tame concealed algebras is closely related to the list of A. Seven of minimal infinite cluster quivers.
From tilted to cluster-tilted algebras of Dynkin type (2005)
Buan, Aslak Bakke, Reiten, Idun
We show how a cluster-tilted algebra of finite representation type is related to the corresponding tilted algebra, in the case of algebras defined over an algebraically closed field.
Clusters and seeds in acyclic cluster algebras (2005)
Buan, Aslak Bakke, Marsh, Robert J., Reiten, Idun, Todorov, Gordana
We show that for cluster algebras associated with finite quivers without oriented cycles (with no coefficients), a seed is determined by its cluster, as conjectured by Fomin and Zelevinsky.We also...
Cluster-tilted algebras of finite representation type (2005)
Buan, Aslak Bakke, Marsh, Robert J., Reiten, Idun
We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation...
Cluster algebras associated with extended Dynkin quivers (2005)
Buan, Aslak Bakke, Reiten, Idun
We show that the mutation class of a finite quiver without oriented cycles is finite if and only is the quiver is either Dynkin, extended Dynkin or has at most two vertices.
Cluster mutation via quiver representations (2004)
Buan, Aslak Bakke, Marsh, Robert J., Reiten, Idun
Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories...
Cluster-tilted algebras (2004)
Buan, Aslak Bakke, Marsh, Robert J., Reiten, Idun
We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory...
Tilting theory and cluster combinatorics (2004)
Buan, Aslak Bakke, Marsh, Robert, Reineke, Markus, Reiten, Idun, Todorov, Gordana
We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a...
Infinite dimensional representations of canonical algebras (2002)
Reiten, Idun, Ringel, Claus Michael
The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to the more general case of modules over concealed canonical algebras. Using...
Noetherian hereditary categories satisfying Serre duality (1999)
Reiten, Idun, Bergh, Michel Van Den
In this paper we classify noetherian hereditary abelian categories satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian...
Tilting Theory and Quasitilted Algebras (1998)
this paper we survey the development of tilting theory, and in particular we discuss quasitilted algebras, a recent outgrowth of tilting theory.