Cluster structures for 2-Calabi-Yau categories and unipotent groups
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In this paper, the authors investigated cluster-tilting objects in triangulated 2-Calabi-Yau and related categories, including pre-projective algebras of non-Dynkin quivers.Abstract:
We investigate cluster-tilting objects (and subcategories) in triangulated 2-Calabi–Yau 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, as special cases, the cluster categories and the stable categories of preprojective algebras of Dynkin graphs. 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 discuss connections with cluster algebras and subcluster algebras related to unipotent groups, in both the Dynkin and non-Dynkin cases.read more
Citations
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Silting mutation in triangulated categories
Takuma Aihara,Osamu Iyama +1 more
TL;DR: In this article, a generalization of the notion of tilting mutation is introduced, called "silting mutation" for the set of subsets of a tilting object that can not be replaced by a new subset.
Book ChapterDOI
Cluster algebras, quiver representations and triangulated categories
TL;DR: In this article, the authors present an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories.
Journal ArticleDOI
Cluster characters for 2-Calabi–Yau triangulated categories
TL;DR: In this article, Caldero et Keller define, for chaque objet L, a fraction rationnelle X(T,L) associated with a caractere amas-basculant T quelconque dans a triangulee 2-Calabi-Yau sur un corps algebriquement clos.
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Cluster tilting for higher Auslander algebras
TL;DR: In this paper, it was shown that the Auslander-Reiten translation functor τ n plays an important role in the study of n-cluster tilting subcategories.
Journal ArticleDOI
Cluster algebras and quantum affine algebras
David Hernandez,Bernard Leclerc +1 more
TL;DR: In this article, the Grothendieck rings of a quantum affine algebra U q ( g ∧ ) of simply laced type were studied using cluster algebras.
References
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Cluster-tilted algebras as trivial extensions
TL;DR: In this paper, the relation-extension algebra of a finite-dimensional algebra C is defined to be the trivial extension C Ext C 2 (DC,C) of C by the C-C-bimodule Ext C2 (C,C).
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Dualizing Complexes, Morita Equivalence and the Derived Picard Group of a Ring
TL;DR: In this paper, it was shown that for the algebra A of upper triangular 2×2 matrices over k, t3 = s, where t, s ∈ DPic(A) are the classes of A*:= Homk(A, k) and A[1] respectively.
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Handbook of Tilting Theory
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