Ian R. Porteous - Böcker

The Clifford algebras of real quadratic forms and their complexifications are studied here in detail, and those parts which are immediately relevant to theoretical physics are seen in the proper broad context. Central to the work is the classification of the conjugation and reversion anti-involutions that arise naturally in the theory. It is of interest that all the classical groups play essential roles in this classification. Other features include detailed sections on conformal groups, the eight-dimensional non-associative Cayley algebra, its automorphism group, the exceptional Lie group G2, and the triality automorphism of Spin 8. The book is designed to be suitable for the last year of an undergraduate course or the first year of a postgraduate course.

The earlier chapter of this self-contained text provide a route from first principles through standard linear and quadratic algebra to geometric algebra, with Clifford's geometric algebras taking pride of place. In parallel with this is an account, also from first principles, of the elementary theory of topological spaces and of continuous and differentiable maps that leads up to the definitions of smooth manifolds and their tangent spaces and of Lie groups and Lie algebras. The calculus is presented as far as possible in basis free form to emphasize its geometrical flavour and its linear algebra content. In this second edition Dr Porteous has taken the opportunity to add a chapter on triality which extends earlier work on the Spin groups in the chapter on Clifford algebras. The details include a number of important transitive group actions and a description of one of the exceptional Lie groups, the group G2. A number of corrections and improvements have also been made. There are many exercises throughout the book and senior undergraduates in mathematics as well as first-year graduate students will continue to find it stimulating and rewarding.