Pure and Applied Mathematics, Marcel Dekker. 235. New York, NY: Marcel Dekker. ix, 401 p. (2001).

Hopf algebras arose in algebraic topology in the work of Heinz Hopf. The paper of {\it J. W. Milnor} and {\it J. C. Moore} [Ann. Math., II. Ser. 81, 211-264 (1965;

Zbl 0163.28202)] on graded Hopf algebras perhaps was the first to attract the attention of algebraists. The first book on the purely algebraic aspects (for the most part ungraded) was that of {\it M. E. Sweedler} [Hopf Algebras, Benjamin, New York (1969;

Zbl 0194.32901)]. Subsequent books of this nature were those of {\it E. Abe} [Hopf Algebras, Cambridge Univ. Press (1980;

Zbl 0476.16008)] and {\it S. Montgomery} [Hopf Algebras and their Actions on Rings, Reg. Conf. Ser. Math. 82, Am. Math. Soc., Providence RI (1993;

Zbl 0793.16029)]. The subject received a huge impetus with the discovery by physicists and mathematicians starting about the mid 1980’s of quantum groups, which are certain noncommutative and noncommutative Hopf algebras. Sweedler’s book is somewhat out of date and Abe’s book is written in a style which is difficult to access, especially for students. The book of Montgomery is an excellent outline, but does not have complete proofs and contains very little directly on quantum groups. Thus there is a need for an introductory book which could be used in a graduate course on Hopf algebras, and the book under review is hence a timely addition to the literature. It can serve as a textbook on the algebraic theory of Hopf algebras. It does not deal with quantum groups, e.g., the quantum Yang-Baxter equation, so that if the instructor wishes to include some quantum groups, he or she would have to add supplementary material. A book which could serve as a textbook for a course on quantum groups is that of {\it C. Kassel} [Quantum Groups, Graduate Texts Math. 155, Springer-Verlag (1995;

Zbl 0808.17003)].
The titles of the seven chapters are: 1. Algebras and coalgebras. 2. Comodules. 3. Special classes of coalgebras. 4. Bialgebras and Hopf algebras. 5. Integrals. 6. Actions and coaction of Hopf algebras. 7. Finite-dimensional Hopf algebras. There are two appendices on category theory language and on $C$-groups and $C$-cogroups.
The presentation is partly categorical and partly concrete algebra, including, for example some Hopf Galois theory and some classification results for finite-dimensional Hopf algebras. Various classes of coalgebras are studied, and investigated for Hopf algebras, with applications given to integrals, Hopf actions and Galois extensions. The idea of duality is extensively used. Some fundamental theorems for finite-dimensional Hopf algebras which are given complete discussions include the Nichols-Zoeller theorem on Hopf subalgebras, the Taft-Wilson theorem on pointed Hopf algebras and the Kac-Zhu theorem on Hopf algebras of prime dimension.
This book is ideal as an up-to-date introduction to the algebraic theory of Hopf algebras. It can be used as a textbook for a one or two semester course. It has exercises of various levels of difficulty scattered in the text, with solutions at the end of each chapter. There are bibliographical notes at the end of each chapter, as well as a bibliography at the end of the book. If such a course wanted to give roughly equal weight to Hopf algebras and to quantum groups, the book under review could be used together with the one of Kassel mentioned above.