Displaying similar documents to “Finite mutation classes of coloured quivers”

Left sections and the left part of an artin algebra

Ibrahim Assem (2009)

Colloquium Mathematicae

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We define a notion of left section in an Auslander-Reiten component, by weakening one of the axioms for sections. We derive a generalisation of the Liu-Skowroński criterion for tilted algebras, then apply our results to describe the Auslander-Reiten components lying in the left part of an artin algebra.

Tame triangular matrix algebras

Zbigniew Leszczyński, Andrzej Skowroński (2000)

Colloquium Mathematicae

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We describe all finite-dimensional algebras A over an algebraically closed field for which the algebra T 2 ( A ) of 2×2 upper triangular matrices over A is of tame representation type. Moreover, the algebras A for which T 2 ( A ) is of polynomial growth (respectively, domestic, of finite representation type) are also characterized.

On domestic algebras of semiregular type

Alicja Jaworska-Pastuszak, Andrzej Skowroński (2013)

Colloquium Mathematicae

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We describe the structure of finite-dimensional algebras of domestic representation type over an algebraically closed field whose Auslander-Reiten quiver consists of generalized standard and semiregular components. Moreover, we prove that this class of algebras contains all special biserial algebras whose Auslander-Reiten quiver consists of semiregular components.

The representation dimension of domestic weakly symmetric algebras

Rafał Bocian, Thorsten Holm, Andrzej Skowroński (2004)

Open Mathematics

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Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional...

On wings of the Auslander-Reiten quivers of selfinjective algebras

Marta Kwiecień, Andrzej Skowroński (2005)

Colloquium Mathematicae

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We give necessary and sufficient conditions for a wing of an Auslander-Reiten quiver of a selfinjective algebra to be the wing of the radical of an indecomposable projective module. Moreover, a characterization of indecomposable Nakayama algebras of Loewy length ≥ 3 is obtained.

Component clusters for acyclic quivers

Sarah Scherotzke (2016)

Colloquium Mathematicae

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The theory of Caldero-Chapoton algebras of Cerulli Irelli, Labardini-Fragoso and Schröer (2015) leads to a refinement of the notions of cluster variables and clusters, via so-called component clusters. We compare component clusters to classical clusters for the cluster algebra of an acyclic quiver. We propose a definition of mutation between component clusters and determine the mutation relations of component clusters for affine quivers. In the case of a wild quiver, we provide bounds...

On selfinjective algebras of tilted type

Andrzej Skowroński, Kunio Yamagata (2015)

Colloquium Mathematicae

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We provide a characterization of all finite-dimensional selfinjective algebras over a field K which are socle equivalent to a prominent class of selfinjective algebras of tilted type.

On Auslander-Reiten translates in functorially finite subcategories and applications

K. Erdmann, D. Madsen, V. Miemietz (2010)

Colloquium Mathematicae

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We consider functorially finite subcategories in module categories over Artin algebras. One main result provides a method, in the setup of bounded derived categories, to compute approximations and the end terms of relative Auslander-Reiten sequences. We also prove an Auslander-Reiten formula for the setting of functorially finite subcategories. Furthermore, we study the category of modules filtered by standard modules for certain quasi-hereditary algebras and we classify precisely when...

Iterated tilted and tilted stably hereditary algebras

Jessica Lévesque (2003)

Colloquium Mathematicae

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We prove that a stably hereditary bound quiver algebra A = KQ/I is iterated tilted if and only if (Q,I) satisfies the clock condition, and that in this case it is of type~Q. Furthermore, A is tilted if and only if (Q,I) does not contain any double-zero.

On the problem of axiomatization of tame representation type

Stanisław Kasjan (2002)

Fundamenta Mathematicae

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Associative algebras of fixed dimension over algebraically closed fields of fixed characteristic are considered. It is proved that the class of algebras of tame representation type is axiomatizable. Moreover, finite axiomatizability of this class is equivalent to the conjecture that the algebras of tame representation type form a Zariski-open subset in the variety of algebras.

Selfinjective algebras of euclidean type with almost regular nonperiodic Auslander-Reiten components

Grzegorz Bobiński, Andrzej Skowroński (2001)

Colloquium Mathematicae

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We give a complete description of finite-dimensional selfinjective algebras of Euclidean tilted type over an algebraically closed field whose all nonperiodic Auslander-Reiten components are almost regular. In particular, we describe the tame selfinjective finite-dimensional algebras whose all nonperiodic Auslander-Reiten components are almost regular and generalized standard.

On self-injective algebras of finite representation type

Marta Błaszkiewicz, Andrzej Skowroński (2012)

Colloquium Mathematicae

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We describe the structure of finite-dimensional self-injective algebras of finite representation type over a field whose stable Auslander-Reiten quiver has a sectional module not lying on a short chain.

On restrictions of indecomposables of tame algebras

R. Bautista, E. Pérez, L. Salmerón (2011)

Colloquium Mathematicae

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We continue the study of ditalgebras, an acronym for "differential tensor algebras", and of their categories of modules. We examine extension/restriction interactions between module categories over a ditalgebra and a proper subditalgebra. As an application, we prove a result on representations of finite-dimensional tame algebras Λ over an algebraically closed field, which gives information on the extension/restriction interaction between module categories of some special algebras Λ₀,...