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
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Tilting theory and cluster combinatorics
TL;DR: In this article, a new category C, called the cluster category, is introduced, 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.
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.
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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.
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On differential graded categories
TL;DR: Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry as discussed by the authors, and they have been extensively studied in recent work by Drinfeld, Dugger-Shipley,..., Toen and Toen-Vaquie.
References
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Book
Representation Theory of Artin Algebras
TL;DR: Artin rings as mentioned in this paper have been used to represent morphisms in the Auslander-Reiten-quiver and the dual transpose and almost split sequences, and they have been shown to be stable equivalence.
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
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.
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.