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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 108 (19 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
, 2009
"... 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 additi ..."
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Cited by 106 (6 self)
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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.
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 56 (11 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
, 2008
"... 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 M ..."
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Cited by 30 (9 self)
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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.
Remarks on noncommutative crepant resolutions of complete intersections
 Adv. Math
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Cluster structures from 2CalabiYau categories with loops
, 2008
"... 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 ..."
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Cited by 12 (1 self)
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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.