Exterior Algebras

Pavan begins the Introduction to his book with a quote from Jean Dieudonné’s article, titled, “The Tragedy of Grassmann,” in which we learn about Hermann Grassmann’s rather dramatic outsider status in 19th century German mathematical circles (he spent his entire teaching career at a secondary school), and the reticence with which others gave credit to him for his development of exterior algebra. Evidently Cauchy was aware of Grassmann’s work, and even used it to great advantage, but failed to credit Grassmann. Although Élie Cartan, in the context of his burgeoning differential geometry, put Grassmann’s work on the mathematical map, Grassmann nonetheless stayed, or rather was kept, under the radar: Engel characterized the work as “Cartanized Grassmann.” Pavan finally offers the following appraisal: “it is only because Cartan’s formalism was better understood that Grassmann’s ideas began to be recognized.”

These important ideas are, of course, the formalism of differential forms, this being where most of us first encounter them (i.e. the exterior calculus, or exterior algebra). Some texts in differential geometry make it a point to develop exterior algebra autonomously and algebraically, and it is certainly the case that in one of my favorite books on algebra Grassmann’s constructs are presented on their own merits, separate from what takes place with differential forms: see p. 715 ff. of Advanced Modern Algebra by J. Rotman. The book under review likewise hits exterior algebra as algebra, per se, and this is in many ways a great virtue, of course: in a sense this is about as abstract and as general as it gets.

Getting down to specifics, the book is presented as “an elementary and progressive” development of Grasmann’s exterior algebra and aims at a broad readership: “students, teachers, as well as other users of algebraic calculations — physicians, computer scientists, or mathematicians …” I imagine that “physicians” is a misprint or a mistranslation, the more reasonable word being “physicists” (seeing that in French, physicist = physicien; Pavan is in Marseille). My doctor is a very smart and cultured man, but I can’t see him having any need for or interest in exterior algebra …

As far as layout and contents go, suffice it to say that Exterior Algebras is well constructed, with statements of definitions, theorems, propositions, and so on, boxed in grey — a very good tactic, I think: it’s pretty sound pedagogy. And the material covered — very thoroughly — is developed at a sound, advanced undergraduate level, and in places somewhat beyond. Solid work habits on the reader’s part will go a very long way. The build-up as regards degrees of difficulty and sophistication is solid, too: among the eleven chapters, in chapter 5 we encounter a very expansive discussion of determinants (if you’ll forgive the cheap pun), while in chapter 9 we’re dealing with Hodge conjugation. Pavan ends with some serious “pure” algebra, viz., endomorphisms of exterior algebras in chapter 10, and \(\bigwedge^2 E\) algebra in chapter 11 (where \(\bigwedge^n E\) is the set of \(n\)-fold wedge products from \(E\), of course).

It is marvelous that Pavan has undertaken to present such a thorough treatment of Grassmann’s work, both because of its intrinsic elegance and because of its utility across mathematics, especially in differential geometry. It is a solid work of scholarship as well as pedagogy.

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