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Clifford Algebras and the Classical Groups

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Overview

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 G[subscript 2], 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.
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Editorial Reviews

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"Porteous' new book is welcome for its coverage and detail and the reliability of its text. Porteous' presentation of the subject matter sets a standard by which others may be judged." Peter R. Law, Mathematical Reviews
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Product Details

Table of Contents

Foreword
1 Linear spaces 1
2 Real and complex algebras 9
3 Exact sequences 18
4 Real quadratic spaces 22
5 The classification of real quadratic spaces 32
6 Anti-involutions of R(n) 43
7 Anti-involutions of C(n) 49
8 Quaternions 57
9 Quaternionic linear spaces 67
10 Anti-involutions of H(n) 72
11 Tensor products of algebras 81
12 Anti-involutions of [superscript 2]K(n) 91
13 The classical groups 100
14 Quadric Grassmannians 110
15 Clifford algebras 123
16 Spin groups 140
17 Conjugation 148
18 2 x 2 Clifford matrices 167
19 The Cayley algebra 178
20 Topological spaces 191
21 Manifolds 202
22 Lie groups 225
23 Conformal groups 245
24 Triality 256
References 285
Index 289
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