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21
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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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
CLUSTER EQUIVALENCE AND GRADED DERIVED EQUIVALENCE
"... Abstract. In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce an invariant of these algebras called cluster equivalence, based on whether their generalized cluster categories are equivalent. We are particularly interested in the question how ..."
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Abstract. In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce an invariant of these algebras called cluster equivalence, based on whether their generalized cluster categories are equivalent. We are particularly interested in the question how much information about an algebra is preserved in its generalized cluster category, or, in other words, how closely two algebras are related if they have equivalent generalized cluster categories. Our approach makes use of the cluster tilting objects in the generalized cluster categories: We first observe that cluster tilting objects in generalized cluster categories are in natural bijection to cluster tilting subcategories of derived categories, and then prove a recognition theorem for the latter. Using this recognition theorem we give a precise criterion when two cluster equivalent algebras are derived equivalent. For a given algebra we further describe all the derived equivalent algebras which have the same canonical cluster tilting object in their generalized cluster category. Finally we show that if two cluster equivalent algebras are not derived equivalent, then
nANGULATED CATEGORIES
"... Abstract. We define nangulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller’s parametrization of pretriangulations extends to prenangulations. We obtain a large class of examples of nangulated categories by considering (n − 2)cluster tilti ..."
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Abstract. We define nangulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller’s parametrization of pretriangulations extends to prenangulations. We obtain a large class of examples of nangulated categories by considering (n − 2)cluster tilting subcategories of triangulated categories which are stable under the (n−2)nd power of the suspension functor. Finally, as an application, we show how nangulated CalabiYau categories yield triangulated CalabiYau categories of higher CalabiYau dimension.
nrepresentationfinite algebras and twisted fractionally Calabi–Yau algebras
, 2011
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Higher dimensional cluster combinatorics and representation theory
"... 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 ..."
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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.
nrepresentation infinite algebras
"... From the viewpoint of higher dimensional AuslanderReiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call nrepresentation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes o ..."
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From the viewpoint of higher dimensional AuslanderReiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call nrepresentation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: npreprojective, npreinjective and nregular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext 1orthogonal families of modules. Moreover we give general constructions of nrepresentation infinite algebras. Applying Minamoto’s theory on Fano algebras in noncommutative algebraic geometry, we describe the category of nregular modules in terms of the corresponding preprojective algebra. Then we introduce nrepresentation tame algebras, and show that the category of nregular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra. This generalizes the classical description of regular modules over tame hereditary algebras. As an application, we show that the representation dimension of an nrepresentation tame algebra is at least n+2.