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16
RIGID MODULES OVER PREPROJECTIVE ALGEBRAS II: THE Kacmoody Case
, 2007
"... Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. We construct many Frobenius subcategories of mod(Λ), which yield categorifications of large classes of cluster algebras. This includes all acyclic cluster algebras. We show that all cluster monomials ..."
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Cited by 40 (7 self)
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Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. We construct many Frobenius subcategories of mod(Λ), which yield categorifications of large classes of cluster algebras. This includes all acyclic cluster algebras. We show that all cluster monomials can be realized as elements of the dual of Lusztig’s semicanonical basis of a universal enveloping algebra U(n), where n is a maximal nilpotent subalgebra of the symmetric KacMoody Lie algebra g associated to the quiver Q.
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 23 (1 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
Derived equivalences from mutations of quivers with potential
 ADVANCES IN MATHEMATICS 226 (2011) 2118–2168
, 2011
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Noncrossing partitions and representations of quivers
 Compos. Math
"... We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yie ..."
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Cited by 9 (0 self)
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We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated to a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extensionclosed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition. 1.
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 8 (7 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.
nrepresentationfinite algebras and nAPR tilting
 6575–6614 (2011) Zbl pre05987996 MR 2833569
"... Abstract. We introduce the notion of nrepresentationfiniteness, generalizing representationfinite hereditary algebras. We establish the procedure of nAPR tilting and show that it preserves nrepresentationfiniteness. We give some combinatorial description of this procedure and use this to compl ..."
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Cited by 6 (5 self)
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Abstract. We introduce the notion of nrepresentationfiniteness, generalizing representationfinite hereditary algebras. We establish the procedure of nAPR tilting and show that it preserves nrepresentationfiniteness. We give some combinatorial description of this procedure and use this to completely describe a class of nrepresentationfinite algebras called “type A”. Contents
The anticyclic operad of moulds
 International Mathematics Research Notices
, 2007
"... A new anticyclic operad Mould is introduced, on spaces of functions in several variables. It is proved that the Dendriform operad is an anticyclic suboperad of this operad. Many operations on the free Mould algebra on one generator are introduced and studied. Under some restrictions, a forgetful map ..."
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Cited by 6 (2 self)
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A new anticyclic operad Mould is introduced, on spaces of functions in several variables. It is proved that the Dendriform operad is an anticyclic suboperad of this operad. Many operations on the free Mould algebra on one generator are introduced and studied. Under some restrictions, a forgetful map from moulds to formal vector fields is then defined. A connection to the theory of tilting modules for quivers of type A is also described. 0
BGPreflection functors and Cluster combinatorics, Journal of Pure and Applied Algebra
"... Abstract. We define BernsteinGelfandPonomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the ”truncated simple reflections ” on the set of almost positive roots Φ≥−1 associated to a finite dimen ..."
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Cited by 5 (2 self)
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Abstract. We define BernsteinGelfandPonomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the ”truncated simple reflections ” on the set of almost positive roots Φ≥−1 associated to a finite dimensional semisimple Lie algebra. Combining with the tilting theory in cluster categories developed in [4], we give a unified interpretation via quiver representations for the generalized associahedra associated to the root systems of all Dynkin types (a simplylaced or nonsimplylaced). This confirms the conjecture 9.1 in [4] in all Dynkin types. Keywords. BGPreflection functor; truncated simple reflection; cluster; cluster categoy; compatibility degree. Mathematics Subject Classification. 16G20, 16G70, 52B11, 17B20. 1.
Higher dimensional cluster combinatorics and representation theory
"... Abstract. Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study i ..."
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Cited by 4 (2 self)
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Abstract. Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection between these two seemingly unrelated subjects. We study triangulations of evendimensional cyclic polytopes and tilting modules for higher Auslander algebras of linearly oriented type A which are summands of the cluster tilting module. We show that such tilting modules correspond bijectively to triangulations. Moreover mutations of tilting modules correspond to bistellar flips of triangulations. For any drepresentation finite algebra we introduce a certain ddimensional cluster category and study its cluster tilting objects. For higher Auslander algebras of linearly oriented type A we obtain a similar correspondence between cluster tilting objects and triangulations of a certain cyclic polytope. Finally we study certain functions on generalized laminations in cyclic polytopes, and show that they satisfy analogues of tropical cluster exchange relations. Moreover we observe that the terms of these exchange relations are closely related to the terms occuring in the mutation of cluster tilting objects. 1.
AuslanderReiten theory revisited
 In Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep
, 2008
"... Abstract. 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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Abstract. 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.