119150095
On indecomposable τ-rigid modules for cluster-tilted algebras of tame type
For a given cluster-tilted algebra $A$ of tame type, it is proved that different indecomposable $\tau$-rigid $A$-modules have different dimension vectors. This is motivated by Fomin-Zelevinsky's denominator conjecture for cluster algebras. As an application, we establish a weak version of the denominator conjecture for cluster algebras of tame type. Namely, we show that different cluster variables have different denominators with respect to a given cluster for a cluster algebra of tame type. Our approach involves Iyama-Yoshino's construction of subfactors of triangulated categories. In particular,we obtain a description of the subfactors of cluster categories of tame type with respect to an indecomposable rigid object, which is of independent interest.
13629643
Cluster algebras I: Foundations
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
14200877
Cluster algebras: Notes for the CDM-03 conference
This is an expanded version of the notes of our lectures given at the conference Current Developments in Mathematics 2003 held at Harvard University on November 21–22, 2003. We present an overview of the main definitions, results and applications of the theory of cluster algebras.
16483706
From triangulated categories to cluster algebras II
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra \(\mathcal{A}\) of finite type can be realized as a Hall algebra, called exceptional Hall algebra, of the cluster category. This realization provides a natural basis for \(\mathcal{A}\). We prove new results and formulate conjectures on ‘good basis’ properties, positivity, denominator theorems and toric degenerations.
16483706
From triangulated categories to cluster algebras II
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra \(\mathcal{A}\) of finite type can be realized as a Hall algebra, called exceptional Hall algebra, of the cluster category. This realization provides a natural basis for \(\mathcal{A}\). We prove new results and formulate conjectures on ‘good basis’ properties, positivity, denominator theorems and toric degenerations.
120690634
The dimension vectors of indecomposable modules of cluster-tilted algebras and the Fomin-Zelevinsky denominators conjecture
The main result of this paper is that any two non-isomorphic indecomposable modules of a cluster-tilted algebra of finite representation type have different dimension vectors. As an application to cluster algebras of Types A,D,E, we give a proof of the Fomin-Zelevinsky denominators conjecture for cluster variables, namely, different cluster variables have different denominators with respect to any given cluster.
119128254
Modules Over Cluster-Tilted Algebras Determined by Their Dimension Vectors
We prove that indecomposable transjective modules over cluster-tilted algebras are uniquely determined by their dimension vectors. Similarly, we prove that for cluster-concealed algebras, rigid modules lifting to rigid objects in the corresponding cluster category are uniquely determined by their dimension vectors. Finally, we apply our results to a conjecture of Fomin and Zelevinsky on denominators of cluster variables.
9371369
Diagrammatic Description of c-vectors and d-vectors of Cluster Algebras of Finite Type
We provide an explicit Dynkin diagrammatic description of the c-vectors and the d-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface realization of cluster algebras for types A_n and D_n, then we apply the folding method to D_{n+1} and A_{2n-1} to obtain types B_n and C_n. Exceptional types are done by direct inspection with the help of a computer algebra software. We also propose a conjecture on the root property of c-vectors for a general cluster algebra.
15318919
Tilting theory and cluster combinatorics
Abstract 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 field. We show that, in the simply laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin–Zelevinsky. Using approximation theory, we investigate the tilting theory of C , showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.
117195350
Triangulated categories in the representation theory of finite dimensional algebras
Preface 1. Triangulated categories 2. Repetitive algebras 3. Tilting theory 4. Piecewise hereditary algebras 5. Trivial extension algebras References Index.
118584112
Tame Algebras and Integral Quadratic Forms
Integral quadratic forms.- Quivers, module categories, subspace categories (Notation, results, some proofs).- Construction of stable separating tubular families.- Tilting functors and tubular extensions (Notation, results, some proofs).- Tubular algebras.- Directed algebras.
119664758
Mutation in triangulated categories and rigid Cohen–Macaulay modules
We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander–Reiten–Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen–Macaulay modules over certain Veronese subrings.
13470097
R T ] 2 9 M ar 2 00 7 Cluster structures for 2-Calabi-Yau categories and unipotent groups A
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 preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related to unipotent groups, both in the Dynkin and non Dynkin case.
12191899
Denominators in cluster algebras of affine type
The Fomin-Zelevinsky Laurent phenomenon states that ev- ery cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for the denominators of cluster vari- ables in cluster algebras of affine type. The formulas are in terms of the dimensions of spaces of homomorphisms in the corresponding cluster category, and hold for any choice of initial cluster.
15851082
Cluster-tilted algebras
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 is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.
15877949
CLUSTER MUTATION VIA QUIVER REPRESENTATIONS
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 of hereditary alge- bras. Using this, we obtain a representation theoretic interpretation of cluster mutation in case of acyclic cluster algebras of finite type.
56415479
Cluster structures from 2-Calabi–Yau categories with loops
We generalise the notion of cluster structures from the work of Buan–Iyama–Reiten–Scott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Hom-finite 2-Calabi–Yau category, the set of maximal rigid objects satisfies these axioms whenever there are no 2-cycles in the quivers of their endomorphism rings. We apply this result to the cluster category of a tube, and show that this category forms a good model for the combinatorics of a type B cluster algebra.
10049600
On triangulated orbit categories
We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by A. Buan, R. Marsh and I. Reiten which appeared in their study with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (closely related to work by Caldero-Chapoton-Schiffler) and a question by H. Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many easy examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel, whose triangulated structure goes back to Auslander-Reiten's work on the representation-theoretic approach to rational singularities.
14822267
Denominators of cluster variables
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 cluster algebra and exceptional indecomposable objects in the cluster category, inducing
a correspondence between clusters and cluster-tilting objects.
Fix a cluster-tilting object T and a corresponding initial cluster. By the Laurent phenomenon,
every cluster variable can be written as a Laurent polynomial in the initial cluster. We give
conditions on T that are equivalent to the fact that the denominator in the reduced form for
every cluster variable in the cluster algebra has exponents given by the dimension vector of the
corresponding module over the endomorphism algebra of T.