Positivity in coefficient-free rank two cluster algebras.

Dupont, Grégoire

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Displaying similar documents to “Positivity in coefficient-free rank two cluster algebras.”

Finite mutation classes of coloured quivers

Hermund André Torkildsen (2011)

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We show that the mutation class of a coloured quiver arising from an m-cluster tilting object associated with a finite-dimensional hereditary algebra H, is finite if and only if H is of finite or tame representation type, or it has at most two simples. This generalizes a result known for cluster categories.

Cluster categories for algebras of global dimension 2 and quivers with potential

Claire Amiot (2009)

Annales de l’institut Fourier

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Let $k$ be a field and $A$ a finite-dimensional $k$ -algebra of global dimension $\leq 2$ . We construct a triangulated category $𝒞_{A}$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$ . When $𝒞_{A}$ is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our...

Finding a cluster-tilting object for a representation finite cluster-tilted algebra

M. A. Bertani-Økland, S. Oppermann, A. Wrålsen (2010)

Colloquium Mathematicae

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We provide a technique to find a cluster-tilting object having a given cluster-tilted algebra as endomorphism ring in the finite type case.

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.

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...

Roots of Nakayama and Auslander-Reiten translations

Helmut Lenzing, Andrzej Skowroński (2000)

Colloquium Mathematicae

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We discuss the roots of the Nakayama and Auslander-Reiten translations in the derived category of coherent sheaves over a weighted projective line. As an application we derive some new results on the structure of selfinjective algebras of canonical 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...

From triangulated categories to cluster algebras II

Philippe Caldero, Bernhard Keller (2006)

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Selfinjective and Simply Connected Algebras.

Otto Bretscher, C. Läser (1981)

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Left-sided quasi-invertible bimodules over Nakayama algebras

Zygmunt Pogorzały (2005)

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Bimodules over triangular Nakayama algebras that give stable equivalences of Morita type are studied here. As a consequence one obtains that every stable equivalence of Morita type between triangular Nakayama algebras is a Morita equivalence.

Morphisms in the category of finite-dimensional absolute valued algebras

Seidon Alsaody (2011)

Colloquium Mathematicae

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This is a study of morphisms in the category of finite-dimensional absolute valued algebras whose codomains have dimension four. We begin by citing and transferring a classification of an equivalent category. Thereafter, we give a complete description of morphisms from one-dimensional algebras, partly via solutions of real polynomials, and a complete, explicit description of morphisms from two-dimensional algebras. We then give an account of the reducibility of the morphisms, and for...

Exponentiality in categories of partial algebras.

Šlapal, Josef (2002)

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