Berlin: Springer (ISBN 978-0-387-72765-3/hbk; 978-0-387-72766-0/ebook). xiv, 273 p. EUR 59.95/net; £ 53.99; SFR 86.00 (2011).
There are several ways that one can describe Hopf algebras. Historically, Hopf algebras arose in the 1940s by Hopf et al. as a tool used to study the cohomology of compact Lie groups and their homogeneous spaces. Hopf algebras can also be viewed as a generalization of group rings, leading to a generalization of Galois extensions to Hopf-Galois extensions.
In this book, Underwood chooses to introduce Hopf algebras in a manner most natural to a reader whose knowledge of algebra does not extend much beyond a first year graduate course. The notion of group naturally generalizes to a group scheme, and in this work Hopf algebras arise as the algebras which represent these functors. This approach makes the notion of comultiplication and antipode very natural: they represent the binary operation and inverse in the group scheme. In this way Hopf algebra axioms such as coassociativity are intuitive. This approach restricts the Hopf algebras to commutative Hopf algebras, and hence would not be the best definition for the reader interested in, say, quantum groups, but for applications to problems in Galois module theory the reader receives an excellent introduction to the topic.
One downside for a person searching for a quick definition of “Hopf algebra” is that the term is not defined until Chapter 4. Nonetheless, the first page of Chapter 4 gives a definition of a commutative Hopf algebra which is independent of the theory developed in the first three chapters. The reader might lack the intuition that would be gained from reading the first three chapters, which are devoted to the development of the Zariski topology, sheaves, and schemes, providing the minimum amount of algebraic geometry necessary to relate Hopf algebras to group valued functors.
The particular focus of this book is on the study of Hopf orders. For an integral domain with field of fractions , a finitely generated -algebra is an -Hopf order in the -Hopf algebra is an -submodule of which is closed under multiplication and the comultiplication (from ) and . The case where is a group ring is studied extensively, in particular where , where is the cyclic group of order . Hopf orders in are completely classified, and additional chapters are devoted to and under the assumption that contains a and root of unity, respectively.
Later chapters give applications of Hopf algebras. Given a Galois extension of local fields with rings of integers and , respectively, and Galois group , one is often interested in a description of as an -module. While is not isomorphic to as -modules unless is tame, there may be an -Hopf order in such that as -modules when is wild. A global result ( a finite extension of ) is the computation of the class group of the Hopf order , defined as a quotient of homomorphisms from the virtual characters of to the idèle group of . An important subgroup of , the Hopf-Swan subgroup, is computed. This is a generalization of the Swan subgroup which arises when . Numerous computations of the Hopf-Swan subgroup are provided.
An interesting aspect of this book is the final chapter “Open Questions and Research Problems”. Many of these questions relate to the classification of Hopf orders in cyclic group rings. Some, such as counting the order bounded group valuations on a -group of order , would be accessible to a student with very little algebra background. Others, such as a general, simple classification of Hopf orders in , , seem quite involved.