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
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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.
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Cluster characters for 2-Calabi–Yau triangulated categories
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Cluster tilting for higher Auslander algebras
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Cluster algebras and quantum affine algebras
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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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TL;DR: In this paper, the authors show that the non-crossing partitions associated with a finite Coxeter group form a lattice, which is a natural bijection with the cluster tilting objects in the associated cluster category.
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Journal ArticleDOI
Periodic Algebras which are Almost Koszul
TL;DR: The preprojective algebra and the trivial extension algebra of a Dynkin quiver (in bipartite orientation) are very close to being a Koszul dual pair of algebras as discussed by the authors.