Fundamentals of Hopf Algebras

Certain topological spaces X come equipped with a continuous product operation which satisfies group-like properties, sometimes up to homotopy, and this map induces a product, that is, a ring structure, on the homology groups of X. This product intertwines the natural morphism induced by the diagonal map of X. With appropriate coefficients, these two operations make the homology of such spaces X the both an algebra and a coalgebra, with the two structures interacting rather nicely. The general notion of Hopf algebra is abstracted from this foundational example. With the expected variations, Hopf algebras have appeared in several areas of mathematics, from algebraic topology to algebraic groups and algebraic geometry.

The book under review is an elementary introduction to the theory of abstract Hopf algebras, with minimal prerequisites. As such, it may be used in a basic one-semester graduate course or for self-study. This new book is a more elementary take on Hopf algebras than the author's previous Introduction to Hopf Algebras, a more advanced approach with an algebraic-geometric flavour. Fundamentals is short and gives more detail than one usually expects, for example in long displayed formulas verifying than certain maps are indeed algebra morphisms, are associative or coassociative, are units or counits. One feels that these long-winded displays just beg for a commutative diagram, but this may be just a question of taste.

That being said, the book is quite friendly, and in its first three chapters develops from scratch the basic theory of algebras, coalgebras, bialgebras, and Hopf algebras. Each chapter has a short list of exercises on the same level as the theory just developed. The last chapter is devoted to three important applications of Hopf algebras, from the quantum Yang-Baxter equations, to the equivalence between the category of Hopf algebras over an algebraically closed field and the category of affine group schemes over that field, and to Hopf orders and Galois modules.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.