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Quadratic Mappings and Clifford Algebras

Jacques Helmstetter and Artibano Micali
Publication Date: 
Number of Pages: 
[Reviewed by
Collin Carbno
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In this book, the authors explore connections between quadratic forms and Clifford algebras and try to set the subject on a solid mathematical foundation. After a classical presentation of quadratic mappings and Clifford algebras over arbitrary rings, interior multiplications are introduced that allow for an effective treatment of the deformations of Clifford algebras. Clifford algebras are then discussed using the concept of the Lipschitz monoid; and the Cartan-Chevalley theory of hyperbolic spaces then becomes available for a precise and effective exploration.

These subjects seemed to flow into each other, moving from special to more general structures, and I found the logical relations in the book strong. The last three chapters, as promised by the authors, explore a wider selection of related topics, such as Graded Morita theory, Graded algebras, Hyperbolic spaces, and Witt rings. Generally the format followed in each section was that of presenting some definitions, then propositions, theorems and wrapping up with some examples. Throughout the book there are some historical sections that give a powerful connection to the development of both mathematics and physics.

Perhaps somewhat unusual is the large selection of exercises; my rough estimate is that one fifth of the book is devoted to exercises. These exercises have a diversity of difficulty and style; many provide extensions of the subject material. For the most part, nothing is developed in the exercises that one needs later in the book, which I found useful for a quick reading.

It appears that effort has been taken to make this into a textbook for learning, in that it reads well, there seems to be no large jumps, and proofs are given in sufficent detail to make following them reasonable. The book has, at the end, a biography, a collection of definitions, and a notation section, which I found handy. An index would have been nice, but generally the definitions section worked as such.

The book comes in a sturdy binding, useful size, shape and set with a clean mathematical font. Overall the book would make an excellent graduate or an advanced undergraduate textbook. The only caveat, as a textbook, is the sheer quantity of material; the last three chapters could be dropped to make for a more manageable course load.

I would have liked to have seen this material expanded to appeal more to the physics community. Still, any physics scholar looking to straighten his/her mathematical understanding of Clifford algebras and related areas should find this a useful book. A mathematical person should be delighted at the overview of applications of the ideas found in the historical sections.

The authors set out, using an algebraic approach, to make a self-contained book requiring a limited set of prerequisites on a deep extensive mathematical subject . In my opinion, they have succeeded delightfully.

Collin Carbno is a specialist in process improvement and methodology. He holds a Master’s of Science Degree in theoretical physics and completed course work for Ph.D. in theoretical physics (relativistic rotating stars) in 1979 at the University of Regina. He has been employed for nearly 30 years in various IT and process work at Saskatchewan Telecommunications and currently holds a Professional Physics Designation from the Canadian Association of Physicists, and the Information System Professional designation from the Canadian Information Process Society.