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Book review

Publisher Kluwer Academic Publishers
Year of publication 1993
Reviewed by S.Rogosin

This volume contains some materials on Clifford Algebras and their Applications. The most of them deals with the ideas, approaches, results and their descriptions given by the great mathematician Marcel Riesz. Here are presented the facsimile reproduction of his notes of the lectures delivered at the University of Maryland, College Park (October 1957 - January 1958), the complete contents of which has not been published up to date. The only some editor's supplements are made in addition to this main part of the book. First of all it is the list of erratum to the M. Riesz's lectures so necessary for this type of the text reproducing. Besides in the book is included the M. Riesz's private lectures on Clifford Algebras dictated to E. Folke Bolinder in April 1959. At last P. Lounesto wrote a historical review on the above mentioned lecture notes as well as about the place of M. Riesz's ideas and results on the subject in modern theory of Clifford Algebras any their Applications. All additional materials have to help the readers in understanding the main text. They also distinguish the bright ideas of M. Riesz remained actual for the specialists until now.

At the beginnig the M. Riesz's lectures were supposed to be in six chapters. But unfortunately the only four of them were really prepared (and included into the book). The last two had to be devoted to the problems of applications (to four dimensional real Lorenz space and so to the special relativity theory, as well as to the theory of spinors and so to the solution of the Dirac equation). Their contents is described in the original preface to the lectures by M. Riesz and partly (the fifth lecture) - in the private lecture dictated by M. Riesz to E.F. Bolinder. Besides all the text show the readers the way in which the last two chapters assumed to be written. We can consider the I-IV Chapters not only as one of possible ground for Clifford Number Theory but also (and mainly) as the preparations to the further applications.

The presented four chapters of Marcel Riesz lecture notes give us the imagination about different sides of the Clifford-like Theory.

The chapter I "Clifforn Numbers" is poorely algebraic. The successive and accurate construction of the Clifford numbers is the base for the further theoretical and applied results. The M. Riesz's approach is closed to that of C. Chavalley. On the modern language the big part of this chapter is retranslated by P. Lounesto in the enclosed review. In this review is also presented interconnections between the original M. Riesz's ideas and the recent approaches to the Clifford algebras.

In the chapter II "Rotations and Reflections" is given some aspects of the geometrical constructions used the Clifford Numbers theory.

Then (chapter III "Canonical Ropresentation of Isometries") are studied isometries in a space over the real field and with a metric of Euclidean or Lorentz signature.

At last in the Chapter IY ("Representation of Isometries by Infinitesimal Transformations and Clifford Bivectors") are continued the preparations of the further applications of the Clifford Algebra approach. It is a thorough investigation on squaring, exponentiation, decomposition, contraction and differentiation of bivectors in Clifford algebras. The private lecture notes dictated by M. Riesz to E.Folke Bolinder are in some sense a successive continuation of this chapter's ideas.

The reviewed book has many attractive sides. There are some reasons for its publication. First of all it is the deep original description of the highly developped theory. It contains many interesting ideas not too fully understanding because they were presented before only in some short publications. The discussed approach had a great influence on the modern state of Clifford Numbers theory - many specialists in this branch of mathematics were and are familiar with this text and are using it in their research. The included into the book review of P. Lounesto helps the readers in understanding the contents of M. Riesz's lectures and shows their real place in modern mathematics.

The other reason for this book appearance is of pedagogical type. Prof M. Rieez paid a great attention to this side of mathematical knowledge. His lecture notes shows how one can introduce the deep abstract ideas in the descriptive understandable form. The lectures dictated by M. Riesz to E.Folke Bolinder are made in quite close manner. But they were delivered to the specialist and as a result they were given more directly and concisely.

At last this book is real History of Mathematics. Its editors made all possible to underline its significance for all who is interesting in history not only mathematical results but also in history of ideas.

The book is intended to the wide audience - from the specialists in Clifford Algebras and their applications until the graduate and postgraduate students in mathematics and physics. It will be useful for those who is close to the problems of pedagogical and historical type in Mathematics.