Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories

Iyama, Osamu

doi:10.1016/j.aim.2006.06.002

Cited by 326 publications

(324 citation statements)

References 17 publications

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“…Section 4 discusses maximal 1-orthogonal modules, which were recently introduced and studied by Iyama [25], [26]. We repeat some of Iyama's results, and then study the functor Hom A (−, T ) associated to a maximal 1-orthogonal module T .…”

Section: Theorem 28 With the Notation Above The Following Holdmentioning

confidence: 99%

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Rigid modules over preprojective algebras

Leclerc²,

2006

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Abstract. Let Λ be a preprojective algebra of simply laced Dynkin type ∆. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring C[N ] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type ∆. As an application we obtain that all cluster monomials of C[N ] belong to the dual semicanonical basis.

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Section: Theorem 28 With the Notation Above The Following Holdmentioning

confidence: 99%

“…In particular, maximal rigid modules are examples of maximal 1-orthogonal modules, which play a role in the higher dimensional Auslander-Reiten theory recently developed by Iyama [25], [26]. This yields a direct link between preprojective algebras and classical tilting theory.…”

Section: A λ-Module M Is Called Rigid Provided Extmentioning

confidence: 99%

Rigid modules over preprojective algebras

Leclerc²,

2006

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“…These functors are instrumental in higher Auslander-Reiten theory, see [Iya07b]. We need the following well-known property.…”

Section: Preliminariesmentioning

confidence: 99%

Higher Auslander Correspondence for Dualizing R-Varieties

Iyama

Jasso

2016

Algebr Represent Theor

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Abstract. Let R be a commutative artinian ring. We extend higher Auslander correspondence from Artin R-algebras of finite representation type to dualizing R-varieties. More precisely, for a positive integer d, we show that a dualizing R-variety is d-abelian if and only if it is a d-Auslander dualizing R-variety if and only if it is equivalent to a d-cluster-tilting subcategory of the category of finitely presented modules over a dualizing R-variety.

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“…By Happel ([8], Corollary 4.5(i)) the Auslander-Reiten quiver AR(D b (kD n )) of the bounded derived category D b (kD n ) is the repetition quiver ZD n . We label the vertices with the coordinate system first introduced by Iyama in [11], Definition 4.2 (cf. Figure 1).…”

Section: A Combinatorial Model For the Cluster Category Of Dynkin Typmentioning

confidence: 99%

“…By Proposition 1.6 in [1] the objects in the cluster category C Dn are either induced by kD n -modules or by shifts of projective modules. We restrict the coordinate system from Iyama's paper [11] on the Auslander-Reiten quiver of the derived category to the Auslander-Reiten quiver AR(mod kD n ) of the module category, cf. Figure 15.…”

Section: Lemma 9 Let D Be a Non-crossing Diagram And Letmentioning

confidence: 99%

Mutation of Torsion Pairs in Cluster Categories of Dynkin Type D

Gratz

2015

Appl Categor Struct

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Abstract. In cluster categories, mutation of torsion pairs provides a generalisation for the mutation of cluster tilting subcategories, which models the combinatorial structure of cluster algebras. In this paper we present a geometric model for mutation of torsion pairs in the cluster category C Dn of Dynkin type Dn. Using a combinatorial model introduced by Fomin and Zelevinsky in [7], subcategories in C Dn correspond to rotationally invariant collections of arcs in a regular 2n-gon, which we call diagrams of Dynkin type Dn. Torsion pairs in C Dn have been classified by Holm, Jørgensen and Rubey in [10]. They correspond to so-called Ptolemy diagrams of Dynkin type Dn, which are diagrams of Dynkin type Dn satisfying a certain combinatorial condition. We define mutation of a diagram X of Dynkin type Dn with respect to a compatible diagram D of Dynkin type Dn consisting of pairwise non-crossing arcs. Such a diagram D partitions the regular 2n-gon into cells and mutation of X with respect to D can be thought of as a rotation of each of these cells. We show that mutation of Ptolemy diagrams of Dynkin type Dn corresponds to mutation of the corresponding torsion pairs in the cluster category of Dynkin type Dn.

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Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories

Cited by 326 publications

References 17 publications

Rigid modules over preprojective algebras

Rigid modules over preprojective algebras

Higher Auslander Correspondence for Dualizing R-Varieties

Mutation of Torsion Pairs in Cluster Categories of Dynkin Type D

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