In September 2000, at the Centre de Recerca Matematica in Barcelona, we
pre- sented a 30-hour Advanced Course on Algebraic Quantum Groups. After
the course, we expanded and smoothed out the material presented in the
lectures and inte- grated it with the background material that we had
prepared for the participants; this volume is the result. As our title
implies, our aim in the course and in this text is to treat selected
algebraic aspects of the subject of quantum groups. Sev- eral of the
words in the previous sentence call for some elaboration. First, we mean
to convey several points by the term 'algebraic' - that we are concerned
with algebraic objects, the quantized analogues of 'classical' algebraic
objects (in contrast, for example, to quantized versions of continuous
function algebras on compact groups); that we are interested in
algebraic aspects of the structure of these objects and their
representations (in contrast, for example, to applications to other
areas of mathematics); and that our tools will be drawn primarily from
noncommutative algebra, representation theory, and algebraic geometry.
Second, the term 'quantum groups' itself. This label is attached to a
large and rapidly diversifying field of mathematics and mathematical
physics, originally launched by developments around 1980 in theoretical
physics and statistical me- chanics. It is a field driven much more by
examples than by axioms, and so resists attempts at concise description
(but see Chapter 1. 1 and the references therein).