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13
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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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 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.
Stable categories of higher preprojective algebras
, 2009
"... Abstract. We show that if an algebra is nrepresentationfinite then its (n + 1)preprojective algebra is selfinjective. In this situation, we show that the stable module category is (n + 1)CalabiYau, and, more precisely, it is the (n+1)Amiot cluster category of the stable nAuslander algebra. F ..."
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Abstract. We show that if an algebra is nrepresentationfinite then its (n + 1)preprojective algebra is selfinjective. In this situation, we show that the stable module category is (n + 1)CalabiYau, and, more precisely, it is the (n+1)Amiot cluster category of the stable nAuslander algebra. Finally we show that if the (n + 1)preprojective algebra is not selfinjective, under certain assumptions (which are always satisfied for n ∈ {1, 2}) the result above still holds for
Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories
, 2009
"... For an Artinian (n − 1)Auslander algebra Λ with global dimension n( ≥ 2), we show that if Λ admits a trivial maximal (n − 1)orthogonal subcategory of mod Λ, then Λ is a Nakayama algebra. Further, for a finitedimensional algebra Λ over an algebraically closed field K, we show that Λ is a basic and ..."
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For an Artinian (n − 1)Auslander algebra Λ with global dimension n( ≥ 2), we show that if Λ admits a trivial maximal (n − 1)orthogonal subcategory of mod Λ, then Λ is a Nakayama algebra. Further, for a finitedimensional algebra Λ over an algebraically closed field K, we show that Λ is a basic and connected (n−1)Auslander algebra Λ with global dimension n( ≥ 2) admitting a trivial maximal (n − 1)orthogonal subcategory of mod Λ if and only if Λ is given by the quiver: β1 β2 β3 βn 1 � 2 � 3 � · · · n + 1 modulo the ideal generated by {βiβi+11 ≤ i ≤ n − 1}. As a consequence, we get that a finitedimensional algebra over an algebraically closed field K is an (n − 1)Auslander algebra with global dimension n( ≥ 2) admitting a trivial maximal (n − 1)orthogonal subcategory if and only if it is a finite direct product of K and Λ as above.
nStrongly Gorenstein Projective, Injective and Flat Modules
, 2009
"... In this paper, we study the relation between mstrongly Gorenstein projective (resp. injective) modules and nstrongly Gorenstein projective (resp. injective) modules whenever m = n, and the homological behavior of nstrongly Gorenstein projective (resp. injective) modules. We introduce the notion ..."
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In this paper, we study the relation between mstrongly Gorenstein projective (resp. injective) modules and nstrongly Gorenstein projective (resp. injective) modules whenever m = n, and the homological behavior of nstrongly Gorenstein projective (resp. injective) modules. We introduce the notion of nstrongly Gorenstein flat modules. Then we study the homological behavior of nstrongly Gorenstein flat modules, and the relation between nstrongly Gorenstein flat modules and nstrongly Gorenstein projective (resp. injective) modules.
From Auslander algebras to tilted algebras
"... For an (n − 1)Auslander algebra Λ with global dimension n ≥ 2, we show that if Λ admits a trivial maximal (n − 1)orthogonal subcategory of mod Λ, then Λ is of finite representation type and the projective dimension or injective dimension of any indecomposable module in mod Λ is at most n − 1. As a ..."
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For an (n − 1)Auslander algebra Λ with global dimension n ≥ 2, we show that if Λ admits a trivial maximal (n − 1)orthogonal subcategory of mod Λ, then Λ is of finite representation type and the projective dimension or injective dimension of any indecomposable module in mod Λ is at most n − 1. As a result, we have that for an Auslander algebra Λ with global dimension 2, if Λ admits a trivial maximal 1orthogonal subcategory of mod Λ, then Λ is a tilted algebra of finite representation type; furthermore, in case there exists a unique simple module in mod Λ with projective dimension 2, then the converse also holds true. 1.
Trivial Maximal 1Orthogonal Subcategories For Auslander’s 1Gorenstein Algebras
, 2009
"... Let Λ be an Auslander’s 1Gorenstein Artinian algebra with global dimension 2. If Λ admits a trivial maximal 1orthogonal subcategory of mod Λ, then for any indecomposable module M ∈ mod Λ, we have that the projective dimension of M is equal to 1 if and only if so is its injective dimension and th ..."
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Let Λ be an Auslander’s 1Gorenstein Artinian algebra with global dimension 2. If Λ admits a trivial maximal 1orthogonal subcategory of mod Λ, then for any indecomposable module M ∈ mod Λ, we have that the projective dimension of M is equal to 1 if and only if so is its injective dimension and that M is injective if the projective dimension of M is equal to 2, which implies that Λ is almost hereditary.
nREPRESENTATIONFINITE ALGEBRAS AND FRACTIONALLY CALABIYAU ALGEBRAS
, 2009
"... In this short paper, we study nrepresentationfinite algebras from the viewpoint of fractionally CalabiYau algebras. We shall show that all nrepresentationfinite algebras are twisted fractionally CalabiYau. We also show that twisted n(ℓ−1)CalabiYau algebras of global dimension n are ℓ nrepr ..."
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In this short paper, we study nrepresentationfinite algebras from the viewpoint of fractionally CalabiYau algebras. We shall show that all nrepresentationfinite algebras are twisted fractionally CalabiYau. We also show that twisted n(ℓ−1)CalabiYau algebras of global dimension n are ℓ nrepresentationfinite for any ℓ> 0. As an application, we give a construction of nrepresentationfinite algebras using the tensor product.