On the representation dimension of finite dimensional algebras

TREPODE, SONIA

Graz

Exposición; Conferencia para la Escuela Doctoral; 2019

Universidad de Graz

Our objective in this talk is to explore the relation between the representa-tion theory of an algebra and its homological invariants. We are in particular interested here in the representation dimension of an algebra, introduced by Auslander, which measures in some way the complexity of the morphisms of the module category.There were several attempts to understand, or compute, this invariant. Spe-cial attention was given to algebras of representation dimension three. Thereason for this interest is two-fold. Firstly, it is related to the finitistic dimen-sion conjecture: Igusa and Todorov have proved that algebras of representationdimension three have a finite finitistic dimension. Secondly, because Auslander?sexpectation was that the representation dimension would measure how far analgebra is from being representation-finite. He proved in the 70´s that representationfinite algebras are characterized as the ones having representation dimension two.Based in on this result and the fact that all the examples known having representation dimension greater than three are wild algebras, there is a standing conjecture that the representation dimension of a tame algebra is at most three.Indeed, while there exist algebras of arbitrary, but finite, representationdimension, most of the best understood classes of algebras have representationdimension three. In this talk we consider several classes of algebras and discuss their representationdimension. We discuss the connection with the finitistic conjecture and the relation withtame algebras. Finally, we consider a class of selfinjective algebras, that is, the algebras which are the orbit algebra of the repetitive algebra of some tilted algebras under the action of an infinite cyclic group of automorphisms and we show that their representation dimensionis less or equal to three.