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357
Cluster Categories for Algebras of Global Dimension 2 and . . .
, 2008
"... Let k be a field and A a finitedimensional kalgebra of global dimension ≤ 2. We construct a triangulated category CA associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When CA is Homfinite, we prove that it is 2CY and endowed with a canonical cluster ..."
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Cited by 127 (10 self)
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Let k be a field and A a finitedimensional kalgebra of global dimension ≤ 2. We construct a triangulated category CA associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When CA is Homfinite, we prove that it is 2CY and endowed with a canonical clustertilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by GeissLeclercSchröer and by BuanIyamaReitenScott. Our results rely on quivers with potential. Namely, we introduce a cluster category C (Q,W) associated to a quiver with potential (Q, W). When it is Jacobifinite we prove that it is endowed with a clustertilting object whose endomorphism algebra is isomorphic
Introduction to Ainfinity algebras and modules
, 1999
"... These are slightly expanded notes of four introductory talks on ..."
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Cited by 123 (5 self)
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These are slightly expanded notes of four introductory talks on
CLUSTER ALGEBRAS, QUIVER REPRESENTATIONS AND TRIANGULATED CATEGORIES
"... Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). I ..."
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Cited by 113 (6 self)
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Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau 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. Contents
Cluster structures for 2CalabiYau categories and unipotent groups
"... Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This c ..."
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Cited by 112 (17 self)
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Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This class of 2CalabiYau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2CalabiYau 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 give applications to cluster algebras and subcluster algebras related
Geometry of the Moment Map for Representations of Quivers
 Compositio Math
, 1999
"... . We study the moment map associated to the cotangent bundle of the space of representations of a quiver, determining when it is flat, and giving a stratification of its MarsdenWeinstein reductions. In order to do this we determine the possible dimension vectors of simple representations of deforme ..."
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Cited by 97 (11 self)
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. We study the moment map associated to the cotangent bundle of the space of representations of a quiver, determining when it is flat, and giving a stratification of its MarsdenWeinstein reductions. In order to do this we determine the possible dimension vectors of simple representations of deformed preprojective algebras. In an appendix we use deformed preprojective algebras to give a simple proof of much of Kac's Theorem on representations of quivers in characteristic zero. Keywords: Quiver, Representation, Moment map, Preprojective algebra Mathematics Subject Classifications (2000): Primary 16G20, 53D20 1. Introduction Let K be an algebraically closed field and let Q be a quiver with vertex set I . Representations of Q of dimension vector ff 2 N I are given by elements of the space Rep(Q; ff) = M a2Q Mat(ff h(a) \Theta ff t(a) ; K); where h(a) and t(a) are the head and tail vertices of an arrow a 2 Q; isomorphism classes correspond to orbits of the group G(ff) = / Y i...
Rigid modules over PREPROJECTIVE ALGEBRAS
, 2005
"... Let Λ be a preprojective algebra of simply laced Dynkin type ∆. We study maximal rigid Λmodules, their endomorphism algebras and a mutation operation on these modules. This leads to a representationtheoretic construction of the cluster algebra structure on the ring C[N] of polynomial functions on ..."
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Cited by 92 (13 self)
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Let Λ be a preprojective algebra of simply laced Dynkin type ∆. We study maximal rigid Λmodules, their endomorphism algebras and a mutation operation on these modules. This leads to a representationtheoretic construction of the cluster algebra structure on the ring C[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type ∆. As an application we obtain that all cluster monomials
Cluster mutation via quiver representations
 Comment. Math. Helv
"... Abstract. Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of ..."
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Cited by 83 (16 self)
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Abstract. Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of cluster mutation in case of acyclic cluster algebras.
A geometric approach to perturbation theory of matrices and matrix pencils. Part II: A stratificationenhanced staircase algorithm
 SIAM J. Matrix Anal. Appl
, 1997
"... Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a wellknown illposed problem. We propose that knowledge of the closure relations, i.e., the stratification, of the orbits and bundles of the various forms may be applied in the staircase algorithm. Here we discuss and ..."
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Cited by 65 (10 self)
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Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a wellknown illposed problem. We propose that knowledge of the closure relations, i.e., the stratification, of the orbits and bundles of the various forms may be applied in the staircase algorithm. Here we discuss and complete the mathematical theory of these relationships and show how they may be applied to the staircase algorithm. This paper is a continuation of our Part I paper on versal deformations, but may also be read independently.