Towards derived equivalence classification of the cluster-tilted algebras of Dynkin type D

Bastian, Janine; Holm, Thorsten; Ladkani, Sefi

doi:10.1016/j.jalgebra.2014.03.034

Cited by 16 publications

(20 citation statements)

References 32 publications

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“…Definition 2.1. [BHL2] Let Γ be an algebra given as a quiver with relations and k a vertex without loops.…”

Section: Preliminariesmentioning

confidence: 99%

“…This in turn happens if and only if the Cartan matrices of the algebras have the same determinant and the same characteristic polynomial of their asymmetry matrices. For Dynkin type D n , Bastian, Holm and Ladkani consider the mutations of quivers which preserve derived equivalences, and get a far reaching derived equivalence classification and suggest standard forms for the derived equivalence classes [BHL2].…”

Section: Introductionmentioning

confidence: 99%

“…For cluster-tilted algebras of type A, since they are gentle algebras, it was settled by Kalck in [Ka]. We use the good mutation defined in [BHL2] to give a direct proof, see Corollary 3.8. For type D, using the description of its quiver by Geng and Peng [GP], Vatne [Va], we get some selfinjective algebras, whose stable categories are equivalent to the singularity categories of each type, see Theorem 4.9.…”

Section: Introductionmentioning

confidence: 99%

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The Singularity Categories of the Cluster-Tilted Algebras of Dynkin Type

Chen

Geng

2014

Algebr Represent Theor

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“…Definition 2.1. [BHL2] Let Γ be an algebra given as a quiver with relations and k a vertex without loops.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Singularity Categories of the Cluster-Tilted Algebras of Dynkin Type

Chen

Geng

2014

Algebr Represent Theor

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“…Definition 2.8 ( [BHL2]). Let A be an algebra given as a quiver with relations and k a vertex without loops.…”

Section: Cluster Categories Are By Definition the Orbit Categoriesmentioning

confidence: 99%

Singularity Categories of some 2-CY-tilted Algebras

2016

Algebr Represent Theor

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Abstract. We define a class of finite-dimensional Jacobian algebras, which are called (simple) polygon-tree algebras, as a generalization of cluster-tilted algebras of type D. They are 2-CYtilted algebras. Using a suitable process of mutations of quivers with potentials (which are also BB-mutations) inducing derived equivalences, and one-pointed (co)extensions which preserve singularity equivalences, we find a connected selfinjective Nakayama algebra whose stable category is equivalent to the singularity category of a simple polygon-tree algebra. Furthermore, we also give a classification of algebras of this kind up to representation type.

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“…This description tells us that the quiver is of one of four possible types. Each type consists of a collection of quivers of mutation type A 'glued' to a 'skeleton' quiver (using the terminology of [3]). The four types are outlined below.…”

Section: Explicit Construction Of Companion Bases In Dynkin Type D Nmentioning

confidence: 99%

Explicit Construction of Companion Bases

Parsons¹

2015

Glasgow Math. J.

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A companion basis for a quiver mutation equivalent to a simplylaced Dynkin quiver is a subset of the associated root system which is a ‫-ޚ‬basis for the integral root lattice with the property that the non-zero inner products of pairs of its elements correspond to the edges in the underlying graph of . It is known in type A (and conjectured for all simply-laced Dynkin cases) that any companion basis can be used to compute the dimension vectors of the finitely generated indecomposable modules over the associated cluster-tilted algebra. Here, we present a procedure for explicitly constructing a companion basis for any quiver of mutation type A or D.2010 Mathematics Subject Classification. 13F60, 05E10, 17B22. Introduction.The cluster algebras of finite type were shown to have a classification by Dynkin diagrams in [12]. We consider the cluster algebra A associated to a simply-laced Dynkin diagram . It follows from the classification result [12, Theorem 1.8] that the exchange matrices of A are skew-symmetric integer matrices with entries not exceeding 1 in absolute value (note that skew-symmetry is preserved under matrix mutation). These exchange matrices can therefore be represented as quivers, which we shall refer to as the quivers of mutation type . Fix such a quiver and denote its set of vertices by 0 . In [16] (originally in [15]), motivated by work in [1], we introduced the notion of a companion basis for as a certain subset of the corresponding root system of type . Specifically, a companion basis for is a subset {γ x : x ∈ 0 } ⊆ whose elements form a ‫-ޚ‬basis for the integral root lattice ‫ޚ‬ and such that for distinct vertices x, y ∈ 0 , (γ x , γ y ) is equal to the number of edges joining x and y in the underlying unoriented graph for .This article continues the study of companion bases initiated in [16]. The focus here is on producing explicit companion bases for quivers. In [16], we already presented a method for finding a companion basis for any given quiver of simply-laced Dynkin mutation type. Indeed, we noted there that a simple system of a root system of simplylaced Dynkin type is a companion basis for any orientation of the associated Dynkin diagram. Furthermore, we introduced a companion basis mutation procedure that, given a companion basis for a quiver of simply-laced Dynkin mutation type, produces a companion basis for any mutation of that quiver. While this, at least in theory, enables us to find a companion basis for any quiver of simply-laced Dynkin mutation type, it has a drawback that can make it difficult to apply in practice. Namely, that it requires us to find a sequence of quiver mutations taking us from a quiver for which we already have a companion basis to the quiver for which we desire to produce one. Here, we

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Towards derived equivalence classification of the cluster-tilted algebras of Dynkin type D

Cited by 16 publications

References 32 publications

The Singularity Categories of the Cluster-Tilted Algebras of Dynkin Type

The Singularity Categories of the Cluster-Tilted Algebras of Dynkin Type

Singularity Categories of some 2-CY-tilted Algebras

Explicit Construction of Companion Bases

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