Is there a fast-and-easy way to define, or even describe, Hopf algebras? Pierre Cartier’s 2006 article, *A Primer of Hopf Algebras*, starts off with a reference to Connes-Kreimer and perturbative quantum field theory, so I figured that a good definition should be available in Connes-Marcolli, *Noncommutative Geometry: Quantum Fields and Motives*, given that Connes is not known for beating around the bush (he visited my graduate school in the early 1980s and I heard him speak). Well, we find on p. 67 that a commutative Hopf algebra over a field *k* of characteristic zero is “a commutative algebra with unit over *k*, endowed with a (not necessarily commutative) coproduct…, a counit…, which are *k*-algebra morphisms, and an antipode which is a *k*-algebra antihomomorphism…” and, to boot, these objects satisfy an assortment of so-called “co-rules.”

All right, what about the book under review? Well, we find on p. 55 that an *R*-Hopf algebra is “a commutative *R*-algebra … that is the representing algebra of an *R*-group scheme … That is [!], an *R*-Hopf algebra is a commutative *R*-algebra … together with *R*-algebra homomorphisms,” which immediately harks back to Connes-Marcolli’s coproduct, counit, and antipode — however, now they’re called comultiplication (fair enough), counit (Aha!), and coinverse (well, well) and the same assortment of (three) conditions is appended. My point is made: these are pretty austere and imposing beasts, and there is no fast-and-easy way to get at them.

So this raises the question of why we’d want to go after them at all: there must be a very good reason, what with Heinz Hopf being such a major pioneer in topology, and Alain Connes being the architect of non-commutative geometry. Well, here’s some additional information from Cartier: “The notion of Hopf algebra emerged slowly from the work of the topologists in the 1940s dealing with the cohomology of compact Lie groups and their homogeneous spaces. To fit the needs of topology, severe restrictions were put on these Hopf algebras, namely existence of a grading, (graded) commutativity, etc…” And: “Hopf algebras were introduced in algebraic geometry by Cartier, Gabriel, Manin, Lazard, Grothendieck and Demazure… with great success. Here Hopf algebras play a dual role: first the (left) invariant differential operators on an algebraic group form a cocommutative Hopf algebra, which coincides with the enveloping algebra of the Lie algebra in characteristic 0, but not in characteristic *p*. Second: the regular functions on an affine algebraic group, under ordinary multiplication, form a commutative Hopf algebra.”

And indeed the fog begins to lift — modulo a good deal of preparation: it’s evident that we’re dealing with a subject that has its childhood in topology, its adolescence in algebraic geometry, and now, still young, flourishes in any number of mathematical areas, including even combinatorics; Cartier makes special mention of a deep structural insight by Gian-Carlo Rota in this connection.

So there is an awful lot going on! Here, again, is Cartier: “These methods have been applied to problems in topology (fundamental group of a space), number theory (symmetries of polylogarithms and multizeta numbers), and more importantly, via the notion of a Feynman diagram, to problems in quantum field theory (the work of Connes and Kreimer).”

With this as the context for Hopf algebras, what about the book under review? It is already clear from the definition sketched above that the author’s orientation is, shall we say, post-Grothendieckian (to coin a horrid neologism). Indeed, Underwood states in the book’s Preface that his work “differs from other texts in that Hopf algebras are developed from notions of topological spaces, sheaves, and representable functors,” and he stresses that “algebraic geometry and category theory [should] provide a smooth transition from modern algebra to Hopf algebras.”

This said, a *caveat *is in order. With the aforementioned definition occurring on p. 55, the preceding pages are dense with non-negotiable stuff, with ring spectra and the Zariski topology starting it all off and sheaves appearing already on p. 23; subsequently, in very short order, we get representable functors and representable group functors, and then group schemes, setting the stage for the main act. Thus it’s probably indicated that the reader not be a novice in algebraic geometry and category theory of the indicated sort. Although Underwood develops everything carefully, it’s still pretty esoteric stuff the first time through. Beyond this, the reader is certainly well-advised to hit the exercises in the book with zeal and commitment, even if he has a decent background in commutative algebra, algebraic geometry and category theory.

In any case, once Hopf algebras have been properly introduced and the whole affair gets off the ground, the focus falls on Hopf orders (cf. p.82: they’re pretty accessible, actually), and presently on, for example, connections to Galois module theory. The last three chapters of the book in point of fact deal with particularly attractive “arithmetical” themes such as class groups of Hopf orders. Underwood ends the book with a discussion of “Open questions and research problems.”

Clearly this is very sexy stuff, and Underwood’s book will make a real impact cutting across a number of putative boundaries — how can it be otherwise when the subject hits topology, algebraic geometry, number theory, quantum field theory, and combinatorics, and probably much else besides?

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.