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On the combinatorics of rigid objects in 2CalabiYau categories
 INT. MATH. RES. NOT. IMRN, (11):ART. ID RNN029
, 2008
"... Given a triangulated 2CalabiYau category C and a clustertilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposab ..."
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Cited by 26 (2 self)
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Given a triangulated 2CalabiYau category C and a clustertilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a clustertilting subcategory T ′ form a basis of the Grothendieck group of T and that, if T and T ′ are related by a mutation, then the indices with respect to T and T ′ are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given clustertilting subcategory T. Conjecturally, these indices coincide with FominZelevinsky’s gvectors.
Cluster characters for cluster categories with infinitedimensional morphism spaces
"... We prove the existence of cluster characters for Hominfinite cluster categories. For this purpose, we introduce a suitable mutationinvariant subcategory of the cluster category. We sketch how to apply our results in order to categorify any skewsymmetric cluster algebra. More applications and a c ..."
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Cited by 25 (1 self)
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We prove the existence of cluster characters for Hominfinite cluster categories. For this purpose, we introduce a suitable mutationinvariant subcategory of the cluster category. We sketch how to apply our results in order to categorify any skewsymmetric cluster algebra. More applications and a comparison to DerksenWeymanZelevinsky’s results will be given in a future paper.
Auslander algebras and initial seeds for cluster algebras
 J. LONDON MATH. SOC
, 2006
"... Let Q be a Dynkin quiver and Π the corresponding set of positive roots. For the preprojective algebra Λ associated to Q we produce a rigid Λmodule IQ with r = Π  pairwise nonisomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of kQ to Λ. If N is ..."
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Cited by 24 (6 self)
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Let Q be a Dynkin quiver and Π the corresponding set of positive roots. For the preprojective algebra Λ associated to Q we produce a rigid Λmodule IQ with r = Π  pairwise nonisomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of kQ to Λ. If N is a maximal unipotent subgroup of a complex simply connected simple Lie group of type Q, then the coordinate ring C[N] is an upper cluster algebra. We show that the elements of the dual semicanonical basis which correspond to the indecomposable direct summands of IQ coincide with certain generalized minors which form an initial cluster for C[N], and that the corresponding exchange matrix of this cluster can be read from the Gabriel quiver of EndΛ(IQ). Finally, we exploit the fact that the categories of injective modules over Λ and over its covering ˜ Λ are triangulated in order to show several interesting identities in the respective stable module categories.
Preprojective algebras and cluster algebras
, 2008
"... We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups. ..."
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Cited by 23 (0 self)
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We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.
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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Cited by 21 (9 self)
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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
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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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