Representation type of an Algebra

CHAIO, LE MEUR, TREPODE

Tokyo

Conferencia; International Conference on Representation of Algebras; 2010

The notion of degree of an irreducible morphism was introduced by S. Liu [SL], in 1992. Using this concept he describes the shapesof the Auslander-Reiten components of an artin algebra of infinite representation type. The degree has also shown to be an useful tool to solve many other problems, such as, to characterize when the composite of two irreducible morphisms is a non zero map in radical cube [CCT]. In this work, [CLMT], we consider finite dimensional algebras over an algebraically closed field. Using a generalization of the covering techniques given in [BG], a full solution to the problem of when the composite of $n$ irreducible morphisms belongs to a greater power of the radical, is given. Moreover, we will show that knowing the degree of a finite number of irreducible morphisms we are able to determine if an algebra is of finite representation type. [BG] K. Bongartz, P. Gabriel. Covering Spaces in Representation-Theory. Inventiones Mathematicae 65, (1982), 331-378. Springer Verlag. [CCT] C. Chaio, F.U. Coelho, S. Trepode. On the composite of two irreducible morphisms in radical cube. Journal of Algebra 312, Issue 2 (2007) 650-667. [CLMT] C. Chaio, P. Le Meur, S. Trepode Degrees of irreducible morphisms and finite-representation type. Preprint (2009) arXiv:0911.2296. [SL] S. Liu. Degrees of irreducible maps and the shapes of Auslander-Reiten quivers. J. London Math. Soc (2) 45, (1992) 32-54.