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 Auslanderâ€™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.