Montroll Memorial Lecture Series in Mathematical Physics. 2. Cambridge: Cambridge Univ. Press. xi, 136 p. £ 35.00; $ 49.95 (1995).

As the author explains in the introduction of this delightful book, the octonion algebra may surely be called a beautiful mathematical entity, which has not been systematically used in physics. This and other nonassociative algebras (other than Lie algebras) are presented along the book with motivations and applications to physics.
Instead of trying to cover a large amount of theory, Okubo concentrates on the octonions and other algebras and constructions related to them and studied, among others, by himself in recent years.
A unique feature of this work is that it presents many different constructions of the real division algebra of octonions or of its split version: from scratch in the introduction, relating it to the Hurwitz problem and representations of Clifford algebras in chapter 3, from irreducible representations of the Lie algebras $G_2$, $su(3)$, $su(2) \oplus su(2)$ and $su(2)$ in chapter 6, from a triple product defined in the Lie algebra $sl(3)$ and from other triple products, axiomatically defined, in chapter 8. Of course, the author does not forget to mention Zorn’s vector matrix algebra and the Cayley-Dickson process.
The introductory chapter 1 is devoted to present the sequence of number systems: real and complex numbers, quaternions and octonions. Chapter 2 deals with the first definitions and properties of nonassociative algebras, the main varieties of Lie, Jordan and alternative algebras are introduced. Chapter 3 presents the Hurwitz algebras (that is to say: the unital composition algebras). A sketch of the proof of the so called Hurwitz theorem (1898), which classifies these algebras, is provided and an application to the instanton solution of the $su(2)$ Yang-Mills field is given. Chapter 4 is devoted to study para-Hurwitz and pseudo-octonion algebras (whose forms are nowadays called Okubo algebras), introduced respectively by the author and {\it H. C. Myung} [J. Algebra 67, 479-490 (1980;

Zbl 0451.17001)] and the author [Hadronic J. 1, 1250-1278 (1978;

Zbl 0417.17011)], and which form a very interesting class of non-unital composition algebras (see, for instance the recent article by {\it J. M. Pérez} and the reviewer [Commun. Algebra 24, 1091-1116 (1996)] and the reference therein). These algebras provide examples of real division algebras, which are recalled in chapter 5, together with several links of the algebras mentioned so far to representations of some real Clifford algebras and a “nonassociative Dirac algebra”.
Chapter 6, based on previous work by the author, explains how to construct several nonassociative algebras from representations of Lie algebras. In particular, different constructions of the octonions are given. Chapter 7 bears the title “Algebra of physical observables”. Jordan algebras, as well as flexible Lie-admissible algebras are quoted here. Chapter 8 is devoted to some triple products introduced by the author [J. Math. Phys 34, 3273-3291 and 3292-3315 (1993;

Zbl 0790.15028 and

Zbl 0790.15029)] and also studied by the reviewer [J. Korean Math. Soc. 33, 183-203 (1996)]. These triple systems are used to provide new constructins of the algebra of octonions and applied to get solutions to the Yang-Baxter equation, among other things. In chapter 9 the problem of how to carry out a nonassociative generalization of the Yang-Mills gauge theory is considered. Finally, there is a final chapter 10 where apologies are given for all the omissions.
Many mathematical structures appear in this book, the needed definitions of the theory of nonassociative algebras (other than Lie algebras) are carefully given and detailed proofs are provided here and there. Hence this book provides a nice introduction to some nonassociative algebras and their applications to physics both for physicists and mathematicians.