Hopf algebras or ‘quantum groups’ are natural generalisations of groups. They have many remarkable properties and, nowadays, they come with a wealth of examples and applications in pure mathematics and mathematical physics.

Most important are the quantum groups Uq(g) modelled on, and in some ways more natural than, the enveloping algebras U(g) of simple Lie algebras g. They provide a natural extension of Lie theory. There are also finite-dimensional quantum groups such as bicrossproduct quantum groups associated to the factorisation of finite groups. Moreover, quantum groups are clearly indicative of a more general ‘noncommutative geometry’ in which coordinate rings are allowed to be noncommutative algebras.

This is a self-contained first introduction to quantum groups as algebraic objects. It should also be useful to someone primarily interested in algebraic groups, knot theory or (on the mathematical physics side) q-deformed physics, integrable systems, or conformal field theory. The only prerequisites are basic algebra and linear algebra. Some exposure to semisimple Lie algebras will also be useful.

The approach is basically that taken in my 1995 textbook, to which the present work can be viewed as a companion ‘primer’ for pure mathematicians. As such it should be a useful complement to that much longer text (which was written for a wide audience including theoretical physicists). In addition, I have included more advanced topics taken from my review on Hopf algebras in braided categories and subsequent research papers given in the Bibliography, notably the ‘braided geometry’ of Uq(g). This is material which may eventually be developed in a sequel volume to the 1995 text.