The idea of representing a complex mathematical object by a simpler
one is as old as mathematics itself. It is particularly useful in classification
problems. For instance, a single linear transformation on a finite dimensional
vector space is very adequately characterised by its reduction to its
rational or its Jordan canonical form. It is now generally accepted that the
representation theory of associative algebras traces its origin to Hamilton’s
description of the complex numbers by pairs of real numbers. During the
1930s, E. Noether gave to the theory its modern setting by interpreting representations
as modules. That allowed the arsenal of techniques developed
for the study of semisimple algebras as well as the language and machinery
of homological algebra and category theory to be applied to representation
theory. Using these, the theory grew rapidly over the past thirty years.