An Introduction to Hopf Algebras [Hardcover]

Only book on Hopf algebras aimed at advanced undergraduates

This book offers a unique self-contained introduction to group and ring theory, and thoroughly treats the concept of the spectrum of a ring and the Zariski topology, helping the reader transition smoothly from basic abstract algebra to Hopf algebras.

Preface.- Some Notation.- 1. The Spectrum of a Ring.-2. The Zariski Topology on the Spectrum.-3. Representable Group Functors.-4. Hopf Algebras. -5. Larson Orders.-6. Formal Group Hopf Orders.-7. Hopf Orders in KC_p.-8. Hopf Orders in KC_{p^2}.-9. Hopf Orders in KC_{p^3}.-10. Hopf Orders and Galois Module Theory.-11. The Class Group of a Hopf Order.-12. Open Questions and Research Problems.-Bibliography.-Index.

From the reviews:

The goal of this book is to introduce readers to the use of Hopf algebras in algebraic number theory and Galois module theory, in particular developing the theory of Hopf orders. & The book concludes with a chapter of open problems, making this text very suitable for a beginning graduate student to work towards the research frontier in this area. & This work very nicely complements the other texts on Hopf algebras and is a welcome addition to the literature.

Jan E. Grabowski, Mathematical Reviews, July, 2013

In this book, Underwood chooses to introduce Hopf algebras in a manner most natural to a reader whose knowledge of algebra does not extend much beyond a first year graduate course.

 Alan Koch, zbMath

The last three chapters of the book in point of fact deal with particularly attractive arithmetical themes such as class groups of Hopf orders. Underwood ends the book with a discussion of Open questions and research problems. Clearly this is very sexy stuff, and Underwoods book will make a real impact cutting across a number of putativl£§