Problems Solved by using degrees of irreducible morphisms

C CHAIO

Massachusets

Conferencia; Maurice Auslander distinguish lectures and international Conference; 2013

The concept of degree of an irreducible morphism was introduced byS. Liu, in 1992. This notion has shown to be a very usefultool to solve many problems. In particular, we are ableto 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.It is well known that $A$ is an artin algebra of finiterepresentation type if and only if there exists a positive integer$n$ such that $Re^{n}(X,Y)=0$ for all $A$-module $X, Y$.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 overan algebraically closed field of finite representation type $A$,we are going to show a new bound $n$ such that $Re^n eq 0$but $Re^{n+1}(X,Y)=0$ for all $X, Y in mbox{mod} A$. This bound isgiven in terms of degrees of irreducible morphisms.We are also going to present some recent developments on degreesof irreducible morphisms.