Clifford algebras. An introduction.

*(English)*Zbl 1235.15025
London Mathematical Society Student Texts 78. Cambridge: Cambridge University Press (ISBN 978-1-107-42219-3/pbk; 978-1-107-09638-7/hbk). vii, 200 p. £ 23.99, $ 39.99/pbk; £ 60.00,/hbk$ 99.00/hbk (2011).

This book came to join some other very good works in the literature of Clifford algebras. It is a clear and concise introduction to Clifford algebras directed to advanced undergraduate and graduate students.

The work it divided into three major parts. The first part is concerned with the background needed to develop the theory of Clifford algebras. Some definitions and results involving the concepts of groups, vector spaces and algebras are recalled, and taking advantage to fix the notation and terminology used in the rest of the text. The second is the main part of the book, where Clifford algebras are constructed over real quadratic spaces. Real Clifford algebras have a much richer structure than the complex Clifford algebras, and this is exploited in detail. All important properties of Clifford algebras are discussed, in particular the periodicity theorems, which are key to their classification. Spinor spaces and spin groups are also discussed. The final part of the book is concerned with applications of Clifford algebras. Some important applications in Physics (as in Maxwell and Dirac equations), generalizations of complex analysis, and differential geometry, are discussed in order to call the reader’s attention and to invite him/her to go deeper in those studies, using references that are suggested at the end. The book has dozens of exercises distributed throughout the text.

In the opinion of this reviewer, the book is well written and is recommended to all those wishing to start a mathematical study of Clifford algebras.

The work it divided into three major parts. The first part is concerned with the background needed to develop the theory of Clifford algebras. Some definitions and results involving the concepts of groups, vector spaces and algebras are recalled, and taking advantage to fix the notation and terminology used in the rest of the text. The second is the main part of the book, where Clifford algebras are constructed over real quadratic spaces. Real Clifford algebras have a much richer structure than the complex Clifford algebras, and this is exploited in detail. All important properties of Clifford algebras are discussed, in particular the periodicity theorems, which are key to their classification. Spinor spaces and spin groups are also discussed. The final part of the book is concerned with applications of Clifford algebras. Some important applications in Physics (as in Maxwell and Dirac equations), generalizations of complex analysis, and differential geometry, are discussed in order to call the reader’s attention and to invite him/her to go deeper in those studies, using references that are suggested at the end. The book has dozens of exercises distributed throughout the text.

In the opinion of this reviewer, the book is well written and is recommended to all those wishing to start a mathematical study of Clifford algebras.

Reviewer: Jayme Vaz (Campinas)