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Stable categories of higher preprojective algebras
, 2009
"... Abstract. We show that if an algebra is n-representation-finite then its (n + 1)-preprojective algebra is self-injective. In this situation, we show that the stable module category is (n + 1)-Calabi-Yau, and, more precisely, it is the (n+1)-Amiot cluster category of the stable n-Auslander algebra. F ..."
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Cited by 21 (9 self)
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Abstract. We show that if an algebra is n-representation-finite then its (n + 1)-preprojective algebra is self-injective. In this situation, we show that the stable module category is (n + 1)-Calabi-Yau, and, more precisely, it is the (n+1)-Amiot cluster category of the stable n-Auslander algebra. Finally we show that if the (n + 1)-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n ∈ {1, 2}) the result above still holds for
n-representation-finite algebras and n-APR tilting
- 6575–6614 (2011) Zbl pre05987996 MR 2833569
"... Abstract. We introduce the notion of n-representation-finiteness, generalizing representation-finite hereditary algebras. We establish the procedure of n-APR tilting and show that it preserves n-representation-finiteness. We give some combinatorial description of this procedure and use this to compl ..."
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Cited by 21 (11 self)
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Abstract. We introduce the notion of n-representation-finiteness, generalizing representation-finite hereditary algebras. We establish the procedure of n-APR tilting and show that it preserves n-representation-finiteness. We give some combinatorial description of this procedure and use this to completely describe a class of n-representation-finite algebras called “type A”. Contents
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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Cited by 21 (6 self)
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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
n-ANGULATED CATEGORIES
"... Abstract. We define n-angulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller’s parametrization of pre-triangulations extends to pre-n-angulations. We obtain a large class of examples of n-angulated categories by considering (n − 2)-cluster tilti ..."
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Cited by 16 (1 self)
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Abstract. We define n-angulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller’s parametrization of pre-triangulations extends to pre-n-angulations. We obtain a large class of examples of n-angulated 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 n-angulated Calabi-Yau categories yield triangulated Calabi-Yau categories of higher Calabi-Yau dimension.
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 finite-dimensional algebra Λ over an algebraically closed field K, we show that Λ is a basic and ..."
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Cited by 6 (3 self)
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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 finite-dimensional 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+1|1 ≤ i ≤ n − 1}. As a consequence, we get that a finite-dimensional 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.
