Clifford Algebras

Clifford algebras, built up from quadratic spaces, have applications in many areas of mathematics, as natural generalizations of complex numbers and the quaternions. They are famously used in proofs of the AtiyahSinger index theorem, to provide double covers (spin groups) of the classical groups and to generalize the Hilbert transform. They also have their place in physics, setting the scene for Maxwell's equations in electromagnetic theory, for the spin of elementary particles and for the Dirac equation. This straightforward introduction to Clifford algebras makes the necessary algebraic background including multilinear algebra, quadratic spaces and finite-dimensional real algebras easily accessible to research students and final-year undergraduates. The author also introduces many applications in mathematics and physics, equipping the reader with Clifford algebras as a working tool in a variety of contexts.

Introduction	p. 1
The algebraic environment	p. 5
Groups and vector spaces	p. 7
Groups	p. 7
Vector spaces	p. 12
Duality of vector spaces	p. 14
Algebras, representations and modules	p. 16
Algebras	p. 16
Group representations	p. 22
The quaternions	p. 24
Representations and modules	p. 28
Module homomorphisms	p. 29
Simple modules	p. 30
Semi-simple modules	p. 32
Multilinear algebra	p. 36
Multilinear mappings	p. 36
Tensor products	p. 38
The trace	p. 42
Alternating mappings and the exterior algebra	p. 44
The symmetric tensor algebra	p. 49
Tensor products of algebras	p. 52
Tensor products of super-algebras	p. 56
Quadratic forms and Clifford algebras	p. 59
Quadratic forms	p. 61
Real quadratic forms	p. 61
Orthogonality	p. 63
Diagonalization	p. 64
Adjoint mappings	p. 68
Isotropy	p. 69
Isometries and the orthogonal group	p. 71
The case d = 2	p. 73
The Cartan-Dieudonné theorem	p. 76
The groups SO(3) and SO(4)	p. 80
Complex quadratic forms	p. 82
Complex inner-product spaces	p. 84
Clifford algebras	p. 86
Clifford algebras	p. 86
Existence	p. 89
Three involutions	p. 91
Centralizers, and the centre	p. 95
Simplicity	p. 96
The trace and quadratic form on A(E, q)	p. 100
The group G(E, q) of invertible elements of A(E, q)	p. 101
Classifying Clifford algebras	p. 104
Frobenius' theorem	p. 104
Clifford algebras A(E, q) with dim E = 2	p. 106
Clifford's theorem	p. 108
Classifying even Clifford algebras	p. 109
Cartan's periodicity law	p. 110
Classifying complex Clifford algebras	p. 113
Representing Clifford algebras	p. 114
Spinors	p. 114
The Clifford algebras A_k, k	p. 117
The algebras B_k,k+i and A_{k, k+1}	p. 119
The algebras A_k+1,k and A_{k +2,k}	p. 120
Clifford algebras A(E, q) with dim E = 3	p. 121
Clifford algebras A(E, q) with dim E = 4	p. 124
Clifford algebras A(E, q) with dim E = 5	p. 131
The algebras A₆, B₇, A₇ and A₈	p. 135
Spin	p. 137
Clifford groups	p. 137
Pin and Spin groups	p. 139
Replacing q by ùq	p. 141
The spin group for odd dimensions	p. 142
Spin groups, for d = 2	p. 142
Spin groups, for d = 3	p. 144
Spin groups, for d = 4	p. 146
The group Spin₅	p. 147
Examples of spin groups for d ≥ 6	p. 148
Table of results	p. 151
Some Applications	p. 153
Some applications to physics	p. 155
Particles with spin 1/2	p. 155
The Dirac operator	p. 156
Maxwell's equations	p. 159
The Dirac equation	p. 161
Clifford analyticity	p. 164
Clifford analyticity	p. 164
Cauchy's integral formula	p. 166
Poisson kernels and the Dirichlet problem	p. 167
The Hilbert transform	p. 169
Augmented Dirac operators	p. 170
Subharmonicity properties	p. 172
The Riesz transform	p. 174
The Dirac operator on a Riemannian manifold	p. 177
Representations of Spin_d and SO(d)	p. 179
Compact Lie groups and their representations	p. 179
Representations of SU(2)	p. 181
Representations of Spind and SO(d) for d ≤ 4	p. 182
Some suggestions for further reading	p. 186
References	p. 191
Glossary	p. 193
Index	p. 197
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