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64
Representation Theory of Artin Algebras
 Studies in Advanced Mathematics
, 1994
"... The representation theory of artin algebras, as we understand it today, is a relatively new area of mathematics, as most of the main developments have occurred ..."
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Cited by 657 (9 self)
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The representation theory of artin algebras, as we understand it today, is a relatively new area of mathematics, as most of the main developments have occurred
Clustertilted algebras are Gorenstein and stably
 CalabiYau, Adv. Math
"... Abstract. We prove that in a 2CalabiYau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its CohenMacaulay modules is 3CalabiYau. We deduce in particular that ..."
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Cited by 150 (16 self)
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Abstract. We prove that in a 2CalabiYau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its CohenMacaulay modules is 3CalabiYau. We deduce in particular that clustertilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [27]. In addition, we prove a general result about relative 3CalabiYau duality over non stable endomorphism rings. This strengthens and generalizes the Extgroup symmetries obtained in [27] for simple modules. Finally, we generalize the results on relative CalabiYau duality from 2CalabiYau to dCalabiYau categories. We show how to produce many examples of dcluster tilted algebras. 1.
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
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
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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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 algebra structures and semicanonical bases for unipotent groups
, 2008
"... Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQmodule M (these are certain preinjective CQmodules), we attach a natural subcategory CM of mod(Λ). We show that CM is a ..."
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Cited by 51 (6 self)
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Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQmodule M (these are certain preinjective CQmodules), we attach a natural subcategory CM of mod(Λ). We show that CM is a
Clustertilted algebras of finite representation type
 J. Algebra
, 2006
"... Abstract. We investigate the clustertilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) clustertilted algebra of finite type is uni ..."
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Cited by 47 (11 self)
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Abstract. We investigate the clustertilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) clustertilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of clustertilted algebras of finite representation type are obtained along the way.
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Cited by 38 (15 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
Minimal Resolutions of Algebras
 J. Algebra
, 1999
"... . A method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself. The method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations which is either homogeneous or admissable (w ..."
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Cited by 30 (0 self)
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. A method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself. The method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations which is either homogeneous or admissable (with some additional finiteness restrictions in the latter case). In particular, it applies to any finite dimensional algebra over an algebraically closed field. The method is illustrated by a number of examples, viz. truncated algebras, monomial algebras and Koszul algebras, with the aim of unifying existing treatments of these in the literature. 1991 Mathematics Subject Classification. Primary: 16E99, 18G10. Secondary: 16D20, 16E40, 16G20, 16W50. 1. Introduction A projective resolution of an algebra , considered as a bimodule over itself, is fundamental in governing the homological properties of the algebra. Such a resolution may be used to compute Hochschild homology and cohomology, to ...