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2015 | Book

Algebras and Coalgebras Read chapter Read first chapter

Fundamentals of Hopf Algebras

Author: Robert G. Underwood

Publisher: Springer International Publishing

Book Series : Universitext

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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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.

Table of Contents

Frontmatter
Chapter 1. Algebras and Coalgebras
Abstract
In this chapter we introduce algebras and coalgebras. We begin by generalizing the construction of the tensor product to define the tensor product of a finite collection of R-modules, where R is a commutative ring with unity.
Robert G. Underwood
Chapter 2. Bialgebras
Abstract
In this chapter we consider bialgebras—vector spaces that are both algebras and coalgebras. We give some basic examples and show that if B is a bialgebra, then B ∘ is a bialgebra. We show that K[x] is a bialgebra in exactly two distinct ways, and so K[x]∘ is a bialgebra in two distinct ways. Consequently, we can multiply linearly recursive sequences in two different ways, namely, the Hadamard product and the Hurwitz product.
Robert G. Underwood
Chapter 3. Hopf Algebras
Abstract
In this chapter we introduce the notion of a Hopf algebra over a field K as a bialgebra with an additional map called the coinverse (or antipode). We discuss some basic features of Hopf algebras and give some initial examples, including the group ring KG. In many respects, KG is the example that is generalized in the concept of Hopf algebra.
Robert G. Underwood
Chapter 4. Applications of Hopf Algebras
Abstract
In this chapter we present three diverse applications of Hopf algebras. Our first application involves almost cocommutative bialgebras and quasitriangular bialgebras. We show that a quastitriangular bialgebra determines a solution to the Quantum Yang–Baxter Equation, and we give details on how to compute quastitriangular structures for certain two-dimensional bialgebras and Hopf algebras. We show that almost cocommutative Hopf algebras generalize Hopf algebras in which the coinverse has order 2. We then define the braid group on three strands (or more simply, the braid group) and show that a quasitriangular structure determines a representation of the braid group.
Robert G. Underwood
Backmatter
Metadata
Title
Fundamentals of Hopf Algebras
Author
Robert G. Underwood
Copyright Year
2015
Electronic ISBN
978-3-319-18991-8
Print ISBN
978-3-319-18990-1
DOI
https://doi.org/10.1007/978-3-319-18991-8

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