An Introduction to Hopf Algebras

1. Front Matter

   Pages i-xiv

   PDF

2. The Spectrum of a Ring

   • Robert G. Underwood

   Pages 1-12

3. The Zariski Topology on the Spectrum

   • Robert G. Underwood

   Pages 13-34

4. Representable Group Functors

   • Robert G. Underwood

   Pages 35-54

5. Hopf Algebras

   • Robert G. Underwood

   Pages 55-94

6. Valuations and Larson Orders

   • Robert G. Underwood

   Pages 95-113

7. Formal Group Hopf Orders

   • Robert G. Underwood

   Pages 115-128

8. Hopf Orders in KC _p

   • Robert G. Underwood

   Pages 129-139

9. Hopf Orders in KC _p ²

   • Robert G. Underwood

   Pages 141-180

10. Hopf Orders in KC _p ³

    • Robert G. Underwood

    Pages 181-194

11. Hopf Orders and Galois Module Theory

    • Robert G. Underwood

    Pages 195-231

12. The Class Group of a Hopf Order

    • Robert G. Underwood

    Pages 233-259

13. Open Questions and Research Problems

    • Robert G. Underwood

    Pages 261-265

14. Back Matter

    Pages 267-273

    PDF

With wide-ranging connections to fields from theoretical physics to computer science, Hopf algebras offer students a glimpse at the applications of abstract mathematics. This book is unique in making this engaging subject accessible to advanced undergraduate and beginning graduate students. After providing a self-contained introduction to group and ring theory, the book thoroughly treats the concept of the spectrum of a ring and the Zariski topology. In this way the student transitions smoothly from basic abstract algebra to Hopf algebras. The importance of Hopf orders is underscored with applications to algebraic number theory, Galois module theory and the theory of formal groups. By the end of the book, readers will be familiar with established results in the field and ready to pose research questions of their own.

• Galois module theory
• Hopf algebras
• Hopf orders
• Larson orders
• Zariski topology

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 Underwood’s book will make a real impact cutting across a number of putative boundaries…"

 —Michael Berg, MAA Reviews

• Dept. Mathematics, Auburn University, Montgomery, Montgomery, USA

  Robert G. Underwood

Robert G. Underwood is an Associate Professor of Mathematics at Auburn University. His research interests include the classification of Hopf algebra orders in group rings and the application of Hopf orders to Galois module theory. Professor Underwood earned his PhD in Mathematics in 1992 from the State University of New York at Albany, and his Masters in Math Education in 1986, also from SUNY Albany.

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