Cluster tilted algebras with a cyclically oriented quiver

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.Comment: 14 pages, 8 figure

On finite dimensional Jacobian Algebras

We show that Jacobian algebras arising from a sphere with $n$-punctures, with $n\geq5$, are finite dimensional algebras. We consider also a family of cyclically oriented quivers and we prove that, for any primitive potential, the associated Jacobian algebra is finite dimensional.Comment: Improvements in the grammar of the article and change the results of the last sectio

Degrees of irreducible morphisms and finite-representation type

We study the degree of irreducible morphisms in any Auslander-Reiten component of a finite dimensional algebra over an algebraically closed field. We give a characterization for an irreducible morphism to have finite left (or right) degree. This is used to prove our main theorem: An algebra is of finite representation type if and only if for every indecomposable projective the inclusion of the radical in the projective has finite right degree, which is equivalent to require that for every indecomposable injective the epimorphism from the injective to its quotient by its socle has finite left degree. We also apply the techniques that we develop: We study when the non-zero composite of a path of $n$ irreducible morphisms between indecomposable modules lies in the $n+1$-th power of the radical; and we study the same problem for sums of such paths when they are sectional, thus proving a generalisation of a pioneer result of Igusa and Todorov on the composite of a sectional path.Comment: 20 page

Covering techniques in Auslander-Reiten theory

Given a finite dimensional algebra over a perfect field the text introduces covering functors over the mesh category of any modulated Auslander-Reiten component of the algebra. This is applied to study the composition of irreducible morphisms between indecomposable modules in relation with the powers of the radical of the module category.Comment: Minor modifications. Final version to appear in the Journal of Pure and Applied Algebr

On the representation dimension of tilted and laura algebras

We prove that the representation dimension of a tilted, or of a strict laura algebra, is at most three.Fil: Assem, Ibrahim. University of Sherbrooke; CanadáFil: Platzeck, Maria Ines. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Trepode, Sonia Elisabet. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentin

Representation Dimension of Cluster Concealed Algebras

We are going to show that the representation dimension of a cluster-concealed algebra B is 3. We compute its representation dimension by showing an explicit Auslander generator for the cluster-tilted algebra.Comment: 15 page