Degrees of irreducible morphisms in standard components

CHAIO CLAUDIA

Sherbrooke

Congreso; Colloquie of non-conmutative algebras; 2008

Universite de Sherbrooke.

The notion of degree of an irreducible morphism, introduced by S. Liu, has shown to be a very helpful tool to solve many problems. Using this concept, he describes the shape of the Auslander-Reiten components for algebras of infinite representation type. It has played an important role in the problem of finding necessary and sufficient conditions for the existence of n irreducible morphisms between indecomposable modules, with non zero composite lying in a greater power of the radical. It has also been a fundamental tool to describe the shape of the Auslander-Reiten components with sectional bypasses. In this work, we consider finite dimensional algebras over an algebraically closed field. We give two criteria to determine whether the degree of an irreducible morphism between modules over a standard component of the Auslander-Reiten quiver, is finite or infinite. We also study the irreducible morphisms of left degree two.