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.