Overview
- 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
Part of the book series: Universitext (UTX)
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Table of contents (4 chapters)
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Front Matter
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Back Matter
Keywords
About this book
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.
Reviews
“The goal of the book under review is to introduce graduate students to some basic results on coalgebras, bialgebras, Hopf algebras, and their applications. The book may be used as the main text or as a supplementary text for a graduate course. … This book should be very useful as a first introduction for someone who wants to learn about Hopf algebras and their applications.” (Jörg Feldvoss, zbMATH 1341.16034, 2016)
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Fundamentals of Hopf Algebras
Authors: Robert G. Underwood
Series Title: Universitext
DOI: https://doi.org/10.1007/978-3-319-18991-8
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2015
Softcover ISBN: 978-3-319-18990-1Published: 25 June 2015
eBook ISBN: 978-3-319-18991-8Published: 10 June 2015
Series ISSN: 0172-5939
Series E-ISSN: 2191-6675
Edition Number: 1
Number of Pages: XIV, 150
Number of Illustrations: 21 b/w illustrations
Topics: Associative Rings and Algebras, Commutative Rings and Algebras, Number Theory, Mathematical Applications in Computer Science