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.

Includes exercises at the end of each chapter designed to reinforce the material Covers topics not typically included in a standard abstract algebra course, including group rings, localizations, absolute values and completions, discriminants and coalgebras, bialgebras and Hopf algebras Serves as a bridge to advanced algebra texts Includes important applications of bialgebras and Hopf algebras to theoretical computer science, knot theory, algebraic geometry and Galois module theory

Autorentext

Robert G. Underwood is a Professor in both the Department of Mathematics and the Department of Computer Science in the College of Sciences at Auburn University at Montgomery. His research in mathematics involves the classification of Hopf orders in group rings and the application of Hopf orders to Galois module theory. His interests in Computer Science include cryptography and automata theory, specifically, algebraic generalizations of the Myhill-Nerode theorem. Professor Underwood is the author of "Fundamentals of Hopf Algebras" and "An Introduction to Hopf Algebras."

Inhalt
Preface.- Notation.- 1. Algebras and Coalgebras.- 2. Bialgebras.- 3. Hopf Algebras.- 4. Applications of Hopf Algebras.- Bibliography.