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
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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.
MonographDOI
Triangulated Categories in the Representation of Finite Dimensional Algebras
TL;DR: The use of triangulated categories in the study of representations of finite-dimensional algebras has been studied extensively in the literature as discussed by the authors, and triangulation is a useful tool in studying tilting processes.
Book
Combinatorics of Coxeter Groups
Anders Björner,Francesco Brenti +1 more
TL;DR: In this paper, the basics of Bruhat order, weak order and reduced words are discussed. But they do not mention the R-polynomials of Kazhdan-Lusztig representations.
Book
Tame Algebras and Integral Quadratic Forms
TL;DR: In this article, the construction of stable separating tubular families and tubular algebras are discussed. But they do not discuss the relation between tubular extensions and directed algesbras.