Cluster structures for 2-Calabi-Yau categories and unipotent groups
TL;DR: 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.
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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.
Abstract: In representation theory of algebras the notion of 'mutation' often plays important roles, and two cases are well known, i.e. 'cluster tilting mutation' and 'exceptional mutation'. In this paper we focus on 'tilting mutation', which has a disadvantage that it is often impossible, i.e. some of summands of a tilting object can not be replaced to get a new tilting object. The aim of this paper is to take away this disadvantage by introducing 'silting mutation' for silting objects as a generalization of 'tilting mutation'. We shall develop a basic theory of silting mutation. In particular, we introduce a partial order on the set of silting objects and establish the relationship with 'silting mutation' by generalizing the theory of Riedtmann-Schofield and Happel-Unger. We show that iterated silting mutation act transitively on the set of silting objects for local, hereditary or canonical algebras. Finally we give a bijection between silting subcategories and certain t-structures.
287 citations
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.
Abstract: This is 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. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.
273 citations
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.
Abstract: Etant donne un objet amas-basculant T quelconque dans une categorie triangulee 2-Calabi-Yau sur un corps algebriquement clos (comme dans le cadre de Keller et Reiten), il est possible de definir, pour chaque objet L, une fraction rationnelle X(T,L), en utilisant une formule proposee par Caldero et Keller. On montre; de plus, que l'application associant X(T,L) a L est un caractere amasse; c'est-a-dire qu'elle verifie une certaine formule de multiplication. Cela permet de prouver qu'elle induit, dans les cas fini et acyclique, une bijection entre objets rigides indecomposables de la categorie amassee et variables d'amas de l'algebre amassee correspondante, confirmant ainsi une conjecture de Caldero et Keller.
218 citations
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.
213 citations
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.
Abstract: Let C be the category of finite-dimensional representations of a quantum affine algebra U q ( g ∧ ) of simply laced type. We introduce certain monoidal subcategories C l ( l ∈ N ) of C , and we study their Grothendieck rings using cluster algebras.
213 citations
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Book•
11 May 2010TL;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.
Abstract: 1. Artin rings 2. Artin algebras 3. Examples of algebras and modules 4. The transpose and the dual 5. Almost split sequences 6. Finite representation type 7. The Auslander-Reiten-quiver 8. Hereditary algebras 9. Short chains and cycles 10. Stable equivalence 11. Modules determining morphisms.
2,044 citations
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.
Abstract: In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
1,942 citations
11 Feb 1988
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.
Abstract: This book is an introduction to the use of triangulated categories in the study of representations of finite-dimensional algebras. In recent years representation theory has been an area of intense research and the author shows that derived categories of finite-dimensional algebras are a useful tool in studying tilting processes. Results on the structure of derived categories of hereditary algebras are used to investigate Dynkin algebras and interated tilted algebras. The author shows how triangulated categories arise naturally in the study of Frobenius categories. The study of trivial extension algebras and repetitive algebras is then developed using the triangulated structure on the stable category of the algebra's module category. With a comprehensive reference section, algebraists and research students in this field will find this an indispensable account of the theory of finite-dimensional algebras.
1,815 citations
Book•
19 Oct 2010
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.
Abstract: I.- The basics.- Bruhat order.- Weak order and reduced words.- Roots, games, and automata.- II.- Kazhdan-Lusztig and R-polynomials.- Kazhdan-Lusztig representations.- Enumeration.- Combinatorial Descriptions.
1,658 citations
Book•
01 Dec 1984
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.
Abstract: Integral quadratic forms.- Quivers, module categories, subspace categories (Notation, results, some proofs).- Construction of stable separating tubular families.- Tilting functors and tubular extensions (Notation, results, some proofs).- Tubular algebras.- Directed algebras.
1,581 citations