On the representation dimension of finite dimensional k-algebras and Auslander's generators

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. The
reason for this interest is two-fold. Firstly, it is related to the finitistic dimen-
sion conjecture: Igusa and Todorov have proved that algebras of representation
dimension three have a finite finitistic dimension. Secondly, because Auslander’s
expectation was that the representation dimension would measure how far an
algebra is from being representation-finite. He proved in the 70's that representation
finite 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, representation
dimension, most of the best understood classes of algebras have representation
dimension three. 

In this talk we consider several classes of algebras and discuss their representation
dimension. We discuss the connection with the finitistic conjecture and the relation with
tame algebras. 

Finally, we present a work in progress with Heily Wagner, Edson Ribeiro Alvares and
Clezio Braga. We consider a class of  algebras having a minimal Auslander's generator and we call 
these algebras representation hereditary algebras. We derive some properties of
these algebras and we discuss some characterizations of them.