The representation dimension of selfijective algebras of tilted type

ASSEM, IBRAHIM; SKOWRONSKI, ANDRZEJ; TREPODE, SONIA

Ciudad de México

Conferencia; ARTA VII/ Advances in Representation Theory of Algebras; 2018

UNAM

The representation dimension of selfinjective algebras oftilted typeSonia TrepodeUniversidad Nacional de Mar del PlataJoint work with Ibrahim Assem and Andrzej Skowro ́nski.Our objective in this talk is to explore the relation between the representa-tion theory of an algebra, or more precisely the shape of its Auslander-Reitencomponents, and its homological invariants. We are in particular interested herein the representation dimension of an algebra, introduced by Auslander, whichmeasures 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, there is a standing conjecture thatthe 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. This is the case, for instance, for algebras obtained by means oftilting, such as tilted algebras, iterated tilted algebras and quasitilted algebras.In ths talk we consider algebras which are the orbit algebra of the repetitivealgebra of some tilted algebra under the action of an infinite cyclic group ofautomorphisms. We prove that the representation dimension of a selfinjectivealgebra of Euclidean or wild tilted type is equal to three, and give an explicitconstruction of an Auslander generator of its module category.We also show that if a connected selfinjective algebra admits an acyclic gener-alised standard Auslander-Reiten component then its representation dimensionis equal to three.