Claudio Procesi’s *Lie Groups: An Approach through Invariants and Representations*, evolved from the locally published Brandeis University Lecture Notes, *A Primer of Invariant Theory*, compiled in 1982 by Giandomenico Boffi, coincidentally the co-author of Threading Homology Through Algebra, reviewed in this venue only a little while ago. Procesi indicated in the *Primer*’s Preface that the lectures’ objective was to “start giving an account of the progress made, since the publication of H. Weyl’s book, *The Classical Groups*, in the invariant theory treated there.” Now, a quarter of a century later, in the Introduction to the book now under review, he states: “…I have constantly drawn inspiration from the book of H. Weyl, *Classical Groups*. On the other hand it would be absurd and quite impossibly to *update* this classic” (italics in the original). These remarks are consistent with the fact that *Lie Groups* weighs in at nearly 600 pages, while the *Primer* had a shade over 200 pages. And obviously Procesi has shifted his focus from invariants *per se* to the titanic if gorgeous subject of Lie groups. But the themes covered retain close ties to their unquestionably classical origins and the explicit acknowledgement of Hermann Weyl’s work is still particularly apposite.

This said, *Lie Groups: An Approach through Invariants and Representations* is a spectacular book. Procesi’s prose is economical and elegant (see the next paragraph for an example of his exposition), the book is a pleasure to read, and the topics covered are perfectly chosen. For Lie groups and Lie algebras, as such, the central chapter is Chapter 4 (at about 40 pages); for representation theory Chapter 8 (also at about 40 pages) is it; and for invariant theory we have Chapter 11 (at over 100 pages). Each of these three chapters can in fact be used as something of a self-contained course in the corresponding subject, modulo a bit of skipping through the rest of the book (or some “mathematical maturity,” to use a phrase that is anything but well-defined). These chapters are supplemented by wonderful treatments of symmetric functions (Ch. 2), algebraic forms (Ch. 3), tensors and tensor symmetry (Chs. 5, 9), and semisimplicity (Chs. 6, 10), as well as discussions of tableaux and standard monomials (Chs. 12, 13), and Hilbert (-Mumford) Theory and binary forms (Chs. 14, 15). What a ride! (And it’s only the beginning, given that Procesi only intends the book for a graduate course.)

Regarding Procesi’s enviable style of exposition (an object lesson in pedagogy, really), here is the promised sample (cf. *loc. cit.* p. 96):

**Lie’s theorem**. Let V be a finite-dimensional vector space over the complex numbers, and let L ⊂ End(V) be a linear solvable Lie algebra. There is a basis of V in which L is formed by upper triangular matrices.

*Proof*. The proof can be carried out by induction on dim L. The essential point is again to prove that there is a nonzero vector v ∈ V which is an eigenvector for L. If we can prove this, then we repeat the argument with L acting on V/Fv. If L is 1-dimensional, then this is the usual fact that a linear operator has a nonzero eigenvector.

We start as in Engel’s Theorem …

These preliminary remarks set the stage perfectly for what follows, smoothing out what might otherwise have been rougher patches (certainly to the neophyte). And the whole book is written in this way, making for remarkable accessibility.

Given that it is published in Springer’s *Universitext* series, and therefore is ostensibly meant as a beginning algebraist’s first real contact with this material, *Lie Groups: An Approach through Invariants and Representations* is probably the best book I’ve ever seen for this purpose. Indeed I can’t imagine a better way to get ready for e.g. Varadarajan’s encyclopedic *Lie Groups, Lie Algebras, and their Representations*, or Serre’s terse *Lie Algebras and Lie Groups*, than by studying this book (and a lot of material would be duplicated, of course). Additionally Procesi’s emphasis in the last parts of the book on Hilbert-Mumford theory would make for a fine prelude to geometric invariant theory, with the proviso that the beginner gird his loins because of the concomitant imperative to learn a large chunk of contemporary algebraic geometry (Springer also publishes Hartshorne’s *Algebraic Geometry* …).

*Lie Groups: An Approach through Invariants and Representations* is one of the best-written books I’ve ever seen.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.