Degrees of irreducible morphisms

CHAIO CLAUDIA

Massachusets

Conferencia; Auslander Conference and International Conference; 2012

The concept of degree of an irreducible morphism was introduced by S. Liu, in 1992. This notion has shown to be a very useful tool to solve many problems. In particular, we are able to determine if a finite dimensional algebra over an algebraically closed field is of finite representation type by computing the degree of a finite number of irreducible morphisms. It is well known that A is an artin algebra of finite representation type if and only if there exists a positive integer n such that rad ^n (X,Y)=0 for all A-modules X, Y. In this case, by the Harada and Sai Lemma we can consider n= 2^m-1 where m is the maximum length of all the indecomposable A-modules. In this talk, for a finite dimensional algebra over an algebraically closed field of finite representation type A, we are going to show a new bound n such that rad^n is not zero but rad^{n+1}(X,Y)=0 for all X, Y in mod A. This bound is given in terms of degrees of irreducible morphisms. We are also going to present some recent developments on degrees of irreducible morphisms.