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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 37 (8 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
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 33 (5 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
Mutation of clustertilting objects and potentials
 Amer. Journal Math. (2008
"... Abstract. We prove that mutation of clustertilting objects in triangulated 2CalabiYau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2CYtilted algebras and Jacobian algebras associated with quivers with potentials. We show that cl ..."
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Cited by 18 (4 self)
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Abstract. We prove that mutation of clustertilting objects in triangulated 2CalabiYau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2CYtilted algebras and Jacobian algebras associated with quivers with potentials. We show that clustertilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3CY algebras. The nearly Morita equivalence for 2CYtilted algebras is shown to hold for the finite length modules over Jacobian algebras.
Cluster tilting for higher Auslander algebras
"... Abstract. The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the ca ..."
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Cited by 15 (5 self)
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Abstract. The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category Mn of preinjectivelike modules obtained by applying τn to injective modules repeatedly. We call a finite dimensional algebra Λ ncomplete if Mn = add M for an ncluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n + 1)complete. This gives an inductive construction of ncomplete algebras. For example, any representationfinite hereditary algebra Λ (1) is 1complete. Hence the Auslander algebra Λ (2) of Λ (1) is 2complete. Moreover, for any n ≥ 1, we have an ncomplete algebra Λ (n) which has an ncluster tilting object M (n) such that Λ (n+1) = End Λ (n)(M (n)). We give the presentation of Λ (n) by a quiver with relations. We apply our results to construct ncluster tilting subcategories of derived categories of ncomplete algebras. Contents 1. Our results 3 1.1. ncluster tilting in module categories 4
Cluster structures from 2CalabiYau categories with loops, preprint
, 2008
"... Abstract. We generalise the notion of cluster structures from the work of BuanIyamaReitenScott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Homfinite 2CalabiYau category, the set of maximal rigid objects satisfies these axioms whenever th ..."
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Abstract. We generalise the notion of cluster structures from the work of BuanIyamaReitenScott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Homfinite 2CalabiYau category, the set of maximal rigid objects satisfies these axioms whenever there are no 2cycles in the quivers of their endomorphism rings.
AuslanderReiten theory revisited
 IN TRENDS IN REPRESENTATION THEORY OF ALGEBRAS AND RELATED TOPICS, EMS SER. CONGR. REP
, 2008
"... We recall several results in AuslanderReiten theory for finitedimensional algebras over fields and orders over complete local rings. Then we introduce ncluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained. ..."
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Cited by 2 (2 self)
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We recall several results in AuslanderReiten theory for finitedimensional algebras over fields and orders over complete local rings. Then we introduce ncluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained.
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"... Abstract. In a recent paper Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal CohenMacaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the fir ..."
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Abstract. In a recent paper Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal CohenMacaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second
VANISHING OF EXT, CLUSTER TILTING MODULES AND FINITE GLOBAL DIMENSION OF ENDOMORPHISM RINGS
"... Abstract. Let R be a CohenMacaulay ring and M a maximal CohenMacaulay Rmodule. Inspired by recent striking work by Iyama, BurbanIyamaKellerReiten and Van den Bergh we study the question of when the endomorphism ring of M has finite global dimension via certain conditions about vanishing of Ext ..."
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Abstract. Let R be a CohenMacaulay ring and M a maximal CohenMacaulay Rmodule. Inspired by recent striking work by Iyama, BurbanIyamaKellerReiten and Van den Bergh we study the question of when the endomorphism ring of M has finite global dimension via certain conditions about vanishing of Ext modules. We are able to strengthen certain results by Iyama on connections between a higher dimension version of Auslander correspondence and existence of noncommutative crepant resolutions. We also recover and extend to positive characteristics a recent Theorem by BurbanIyamaKellerReiten on clustertilting objects in the category of maximal CohenMacaulay modules over reduced 1dimensional hypersurfaces. 1.
INTRODUCTION TO CLUSTER TILTING IN 2CALABIYAU CATEGORIES
"... Abstract. Cluster tilting theory reveals combinatorial structure of 2CalabiYau triangulated cate ..."
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Abstract. Cluster tilting theory reveals combinatorial structure of 2CalabiYau triangulated cate