Finding a cluster-tilting object for a representation finite cluster-tilted algebra (original) (raw)

Cluster-tilted algebras of finite representation type

Journal of Algebra, 2006

We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted algebras of finite representation type are obtained along the way.

Constructing tilted algebras from cluster-tilted algebras

Any cluster-tilted algebra is the relation extension of a tilted algebra. Given the distribution of a cluster-tilting object in the Auslander-Reiten quiver of the cluster category, we present a method to construct all tilted algebras whose relation extension is the endomorphism ring of this cluster-tilting object.

Cluster Tilted Algebras with a Cyclically Oriented Quiver

Communications in Algebra, 2013

In association with a finite dimensional algebra A of global dimension two, we consider the endomorphism algebra of A, viewed as an object in the triangulated hull of the orbit category of the bounded derived category, in the sense of Amiot. We characterize the algebras A of global dimension two such that its endomorphism algebra is isomorphic to a cluster-tilted algebra with a cyclically oriented quiver. Furthermore, in the case that the cluster tilted algebra with a cyclically oriented quiver is of Dynkin or extended Dynkin type then A is derived equivalent to a hereditary algebra of the same type.

On the radical of Cluster tilted algebras

2020

We determine the minimal lower bound nnn, with ngeq1n \geq 1ngeq1, where the nnn-th power of the radical of the module category of a representation-finite cluster tilted algebra vanishes. We give such a bound in terms of the number of vertices of the underline quiver. Consequently, we get the nilpotency index of the radical of the module category for representation-finite self-injective cluster tilted algebras. We also study the non-zero composition of mmm, mge2m \ge 2mge2, irreducible morphisms between indecomposable modules in representation-finite cluster tilted algebras lying in the (m+1)(m+1)(m+1)-th power of the radical of their module category.

Quivers with Relations and Cluster Tilted Algebras

Algebras and Representation Theory, 2006

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of these results, we answer several conjectures on the connection between cluster algebras and quiver representations.

M-Cluster Tilted Algebras of Type

Communications in Algebra, 2018

In this paper, we characterize all the nite dimensional algebras that are mcluster tilted algebras of type A. We show that these algebras are gentle and we give an explicit description of their quivers with relations.

m-cluster tilted algebras of Euclidean type

Journal of Algebra

We consider m-cluster tilted algebras arising from quivers of Euclidean type and we give necessary and sufficient conditions for those algebras to be representation finite. For the caseÃ, using the geometric realization, we get a description of representation finite type in terms of (m + 2)-angulations. We establish which m-cluster tilted algebras arise at the same time from quivers of typeÃ and A. Finally, we characterize representation infinite m-cluster tilted algebras arising from a quiver of typeÃ, as m-relations extensions of some iterated tilted algebra of typeÃ. Dedicated to José Antonio de la Peña on the occasion of his 60th birthday

On the Representation Dimension of Tame Cluster Tilted Algebras

2017

The aim of this work is to study the representation dimension of cluster tilted algebras. We prove that the weak representation dimension of tame cluster tilted algebras is equal to three. We construct a generator module that reaches the weak representation dimension, unfortunately this module is not always a cogenerator. We show for which algebras this module gives the representation dimension.

On representation dimension of tame cluster tilted algebras

2017

Strongness of companion bases for cluster-tilted algebras of finite type

Proceedings of the American Mathematical Society, 2018

For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e. gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a cluster-tilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons's conjecture.

Finding a cluster-tilting object for a representation finite cluster-tilted algebra (original) (raw)

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