“In 1914, E. Cartan posed the problem to find all irreducible real Lie algebras.” This tantalizing and evocative sentence launches the back-cover description of the contents of Lectures on Real Semisimple Lie Algebras and Their Representations, the book under review, whose corresponding objective is to serve as an “exposition of the theory of finite dimensional representations of real semisimple Lie algebras.”
What the author, Arkady Onishchik, of Yaroslavl State University in Russia, presents the reader with is a compact discourse on the central theme of indicated representation theory fitted into nine chapters spanning less than eighty pages, the size of many a research article. This suggests, of course, that the author covers this beautiful material at a clip, and it implies the prerequisite that the reader not be a novice in the field. To wit, Chapter, or §,1, “Preliminaries,” starts off: “In this section, some necessary facts from the theory of Lie groups and Lie algebras are formulated without proofs … see e.g. [Helgason, Knapp (Lie Groups Beyond an Introduction), Serre (not his Lie Algebras and Lie Groups but his Complex Semisimple Lie Algebras)] …” This book is obviously not for the raw beginner.
In point of fact, the contents of this opening chapter comprise in and of themselves a roadmap for a very solid first course in Lie theory, specifically emphasizing compact semisimple Lie algebras. It also includes a very nice discussion of Weyl chambers and addresses Cartan matrices. All this before going on to the subject of representations of Lie algebras —in less than ten pages; caveat lector.
Then, starting with §2, it’s off to the races: complexification, real forms and involutive automorphisms, Cartan decomposition and maximal compact subgroups (staples of the whole subject of Lie theory, of course). Next comes the analysis of hom’s and involutions of complex semisimple Lie algebras, and then, climactically (we’re now at §§ 7,8), “Inclusions between real forms under an irreducible representation” and “Real representations of real semisimple Lie algebras.” The book closes with an appendix on Satake diagrams. Quite a tour de force, and the reader should certainly bear this in mind: dabbling in or browsing through the book under review is not what it was meant for, and the reader should know this going in.
On the other hand, even though Onishchik pitches pretty hard, he doesn’t throw any spitballs. The book is eminently readable, replete with a lot of examples sprinkled liberally throughout, theorems (and propositions and lemmas) clearly stated and properly proved, without asking the reader to go to outside sources every second or third page, the book’s compactness notwithstanding. Within its preset limitations Lectures on Real Semisimple Lie Algebras and Their Representations is essentially self-contained. This is indeed reasonable, as this streamlined book does spring, after all, from an advanced lecture series (the author lectured on this material in Brno and in Vienna in 2001 and 2002, respectively).
Lectures on Real Semisimple Lie Algebras and Their Representations is a wonderful contribution to the literature.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.