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. Prerequisites for this book include the standard material on groups, rings, modules, algebraic field extensions, finite fields, and linearly recursive sequences taught in undergraduate courses on these subjects. Otherwise it has been the author’s intention to make the book as self-contained as possible. Most proofs are given in detail and for those that are omitted, as they are beyond the scope of the book, references are provided.
The book consists of four chapters. In Chapter 1 algebras and coalgebras over a field are introduced. It is proved that the linear dual of any coalgebra is an algebra and that the finite dual of any algebra is a coalgebra. Moreover, the finite dual of a polynomial algebra in one variable over a field is identified with the set of all linearly recursive sequences over this field.
Chapter 2 is devoted to bialgebras which are vector spaces that are both algebras and coalgebras satisfying a compatibility condition. The author proves that the finite dual of a bialgebra is a bialgebra. It is also shown that the polynomial algebra in one variable admits exactly two bialgebra structures. Consequently, the same holds for its finite dual, and thus linearly recursive sequences can be multiplied in two ways which correspond to the Hadamard product and the Hurwitz product. The second chapter also contains an application of bialgebras to formal languages, finite automata, and the classical Myhill-Nerode theorem. The latter is generalized in an algebraic setting where a certain function plays the role of the language and a certain bialgebra, the so-called Myhill-Nerode bialgebra, plays the role of the finite automaton that accepts the language. Several examples of Myhill-Nerode bialgebras are provided. In the last section regular sequences are introduced as generalizations of linearly recursive sequences over finite fields.
In Chapter 3 the author discusses Hopf algebras which are bialgebras with an additional map, the so-called antipode or coinverse, satisfying a certain condition. Several examples of Hopf algebras are given and some of their properties are studied. Furthermore, integrals of Hopf algebras and the concept of a Hopf module are defined. Then the Fundamental Theorem for Hopf modules is stated, and it is proved in a special case which implies that integrals of finite-dimensional Hopf algebras are unique up to a scalar. Hopf algebras over commutative rings are considered and Hopf orders in a group algebra are defined. Finally, a certain family of Hopf orders in the group algebra of the cyclic group of prime order is constructed which plays a role in the last section of Chapter 4 which deals with generalizations of Galois extensions to Hopf-Galois extensions.
Chapter 4 consists of three applications of Hopf algebras, namely, quasitriangular Hopf algebras, solutions of the quantum Yang-Baxter equation, and representations of the braid group; coordinate algebras of affine algebraic groups; and a generalization of Galois extensions to rings of integers using Hopf algebras. Some of the applications discussed in the last chapter are further developed in the author’s previous book [An introduction to Hopf algebras. Berlin: Springer (2011; Zbl 1234.16022)].
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. In Chapters 2-4 also questions for further study are provided. This book should be very useful as a first introduction for someone who wants to learn about Hopf algebras and their applications.