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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 15 (4 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.
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.
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.
Mutation in triangulated . . .
, 2007
"... We introduce the notion of mutation on the set of ncluster tilting subcategories in a triangulated category with AuslanderReitenSerre duality. Using this idea, we are able to obtain the complete classifications of rigid CohenMacaulay modules over certain Veronese subrings. ..."
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We introduce the notion of mutation on the set of ncluster tilting subcategories in a triangulated category with AuslanderReitenSerre duality. Using this idea, we are able to obtain the complete classifications of rigid CohenMacaulay modules over certain Veronese subrings.