Quivers with relations arising from clusters (A_n case)
TLDR
In this article, the denominator theorem of Fomin and Zelevinsky was generalized to any cluster algebra and an algebraic realization and a geometric realization of Cat_C were given.Abstract:
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A_n. We associate to each cluster C of U an abelian category Cat_C such that the indecomposable objects of Cat_C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of Cat_C. Then, we generalize the ``denominator Theorem'' of Fomin and Zelevinsky to any cluster.read more
Citations
More filters
Journal Article
On triangulated orbit categories.
TL;DR: In this paper, it was shown that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is triangulated.
Posted Content
Tilting theory and cluster combinatorics
TL;DR: In this paper, a new category C, called the cluster category, which is obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field, is introduced.
Journal ArticleDOI
Mutation in triangulated categories and rigid Cohen–Macaulay modules
Osamu Iyama,Yuji Yoshino +1 more
TL;DR: In this paper, the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality was introduced.
Journal ArticleDOI
Cluster algebras as Hall algebras of quiver representations
TL;DR: In this paper, it was shown that some cluster algebras of type ADE can be recovered from the data of the corresponding quiver representation category, and also provided some explicit formulas for cluster variables.
Journal ArticleDOI
Cluster categories for algebras of global dimension 2 and quivers with potential
TL;DR: In this article, a triangulee C A associee a A, which is triangle-equivalente a la categorie amassee C A si A est hereditaire, is introduced.
References
More filters
Journal ArticleDOI
Cluster algebras I: Foundations
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this article, a new class of commutative algebras was proposed for dual canonical bases and total positivity in semisimple groups. But the study of the algebraic framework is not yet complete.
Journal ArticleDOI
Representation Theory of Artin Algebras I
TL;DR: The first of a series of papers dealing with the representation theory of artin algebras is presented in this paper, where the main purpose is to develop terminology and background material which will be used in the rest of the papers in the series.
Journal ArticleDOI
Cluster algebras II: Finite type classification
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this paper, a complete classification of cluster algebras of finite type is presented, i.e., those with finitely many clusters, which is identical to the Cartan-Killing classification of semisimple Lie algebases and finite root systems.
Posted Content
Tilting theory and cluster combinatorics
TL;DR: In this paper, a new category C, called the cluster category, which is obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field, is introduced.
Journal ArticleDOI
Y-systems and generalized associahedra
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this paper, a simplicial complex A(b) is constructed for an arbitrary finite root system D, which is the face complex of the ordinary associahedron, whereas in type B it produces the Bott-Taubes polytope or cyclohedron.