This text aims to provide graduate students with a self-contained
introduction to topics that are at the forefront of modern algebra,
namely, coalgebras, bialgebras and Hopf algebras. The last chapter
(Chapter 4) discusses several applications of Hopf algebras, some of
which are further developed in the author's 2011 publication, An
Introduction to Hopf Algebras. The book may be used as the main text or
as a supplementary text for a graduate algebra course. Prerequisites for
this text include standard material on groups, rings, modules, algebraic
extension fields, finite fields and linearly recursive sequences.
The book consists of four chapters. Chapter 1 introduces algebras and
coalgebras over a field K; Chapter 2 treats bialgebras; Chapter 3
discusses Hopf algebras and Chapter 4 consists of three applications of
Hopf algebras. Each chapter begins with a short overview and ends with a
collection of exercises which are designed to review and reinforce the
material. Exercises range from straightforward applications of the
theory to problems that are devised to challenge the reader. Questions
for further study are provided after selected exercises. Most proofs are
given in detail, though a few proofs are omitted since they are beyond
the scope of this book.