Clifford Algebras and the Classical Groups

Priekinis viršelis
Cambridge University Press, 1995-10-05 - 295 psl.
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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Turinys

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 Antiinvolutions of Rn
43
7 Antiinvolutions of Cn
49
8 Quaternions
57
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

9 Quaternionic linear spaces
67
10 Antiinvolutions of Hn
72
11 Tensor products of algebras
81
12 Antiinvolutions of 2Kn
91
13 The classical groups
100
14 Quadric Grassmannians
110
23 Conformal groups
245
24 Triality
256
References
285
Index
289
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