An Introduction to Hopf Algebras

  • Robert G. Underwood

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Robert G. Underwood
    Pages 1-12
  3. Robert G. Underwood
    Pages 13-34
  4. Robert G. Underwood
    Pages 35-54
  5. Robert G. Underwood
    Pages 55-94
  6. Robert G. Underwood
    Pages 95-113
  7. Robert G. Underwood
    Pages 115-128
  8. Robert G. Underwood
    Pages 129-139
  9. Robert G. Underwood
    Pages 141-180
  10. Robert G. Underwood
    Pages 181-194
  11. Robert G. Underwood
    Pages 195-231
  12. Robert G. Underwood
    Pages 233-259
  13. Robert G. Underwood
    Pages 261-265
  14. Back Matter
    Pages 267-273

About this book

Introduction

The study of Hopf algebras spans many fields in mathematics including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups and has wide-ranging  connections to fields from theoretical physics to computer science. This text is unique in making this engaging subject accessible to advanced graduate and beginning graduate students and focuses on applications of Hopf algebras to algebraic number theory and Galois  module theory, providing a smooth transition from modern algebra to Hopf algebras.

After providing an introduction to the spectrum of a ring and the Zariski topology, the text treats presheaves, sheaves, and representable group functors.  In this way the student transitions smoothly from basic algebraic geometry 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.

An exercise set is included in each of twelve chapters with questions ranging in difficulty. Open problems and research questions are presented in the last chapter. Prerequisites include an understanding of the  material on groups, rings, and fields normally covered in a basic course in modern algebra.

Keywords

Galois module theory Hopf algebras Hopf orders Larson orders Zariski topology

Authors and affiliations

  • Robert G. Underwood
    • 1
  1. 1.Dept. MathematicsAuburn University, MontgomeryMontgomeryUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-387-72766-0
  • Copyright Information Springer Science+Business Media, LLC 2011
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-387-72765-3
  • Online ISBN 978-0-387-72766-0
  • About this book