Degrees and compositions in Coherent almost cyclic Auslander-Reiten components

CHAIO CLAUDIA AND MALICKI PIOTR

Bogotá

Conferencia; 2nd International Colloquium of Representation of algebras, Alexander Sadavskij; 2016

Universidad Naconal de Colombia

We consider $A$ an artin algebra and $\mbox{mod}\,A$ the category of finitely generated left $A$-modules.One of the aims of the representation theory of artin algebras is to study in which power of the radical of $\mbox{mod}\,A$ belongs a givencomposition of irreducible morphisms.In 1992, S. Liu introduced the notion of degree of an irreducible morphism in order to study the above problem.This notion has shown to be a usefultool to solve many problems. In particular,to determine if a finite dimensional algebra over an algebraicallyclosed field is of finite representation type by computing the degreeof a finite number of irreducible morphisms.In the last years, the theory of degrees and compositions of irreducible morphisms has been developed mostly for finite dimensional algebras over an algebraically closed field.In this work, we deal with artin algebras. We give a characterization to decide if the degrees of irreducible morphisms between indecomposable modules in generalized standard coherent almost cyclic Auslander-Reiten components are finite.We also determine when the composition of $n$ irreducible morphisms between indecomposable modules in quasi-tubesbelongs to the $n+1$ power of the radical of their module category. A quasi-tube is a component where the stable part is a stable tube.Algebras having Auslander-Reiten quivers with quasi-tubes are for example the selfinjective algebras and generalized multicoil algebras.