Clifford algebras : an introduction /

"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 Atiyah-Singer index theorem, to provide double covers (spin groups) of the classical gro...

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Bibliographic Details
Main Author:	Garling, D. J. H.
Format:	Book
Language:	English
Published:	Cambridge ; New York : Cambridge University Press, 2011.
Series:	London Mathematical Society student texts ; 78.
Subjects:	Clifford algebras. Clifford, Álgebras de.

Table of Contents:

pt. 1. The algebraic environment
1. Groups and vector spaces
1.1. Groups
1.2. Vector spaces
1.3. Duality of vector spaces
2. Algebras, representations and modules
2.1. Algebras
2.2. Group representations
2.3. quaternions
2.4. Representations and modules
2.5. Module homomorphisms
2.6. Simple modules
2.7. Semi-simple modules
3. Multilinear algebra
3.1. Multilinear mappings
3.2. Tensor products
3.3. trace
3.4. Alternating mappings and the exterior algebra
3.5. symmetric tensor algebra
3.6. Tensor products of algebras
3.7. Tensor products of super-algebras
pt. Two Quadratic forms and Clifford algebras
4. Quadratic forms
4.1. Real quadratic forms
4.2. Orthogonality
4.3. Diagonalization
4.4. Adjoint mappings
4.5. Isotropy
4.6. Isometries and the orthogonal group
4.7. case d = 2
4.8. Cartan-Dieudonne theorem
4.9. groups SO(3) and SO(4)
4.10. Complex quadratic forms
4.11. Complex inner-product spaces
5. Clifford algebras
5.1. Clifford algebras
5.2. Existence
5.3. Three involutions
5.4. Centralizers, and the centre
5.5. Simplicity
5.6. trace and quadratic form on A(E, q)
5.7. group G(E, q) of invertible elements of A(E, q)
6. Classifying Clifford algebras
6.1. Frobenius' theorem
6.2. Clifford algabras A(E, q) with dim E = 2
6.3. Clifford's theorem
6.4. Classifying even Clifford algebras
8.5. Cartan's periodicity law
6.6. Classifying complex Clifford algebras
7. Representing Clifford algebras
7.1. Spinors
7.2. Clifford algebras Ak,k
7.3. algebras Bk,k+1 and Ak,k+1
7.4. algebras Ak + 1,k and Ak+2,k
7.5. Clifford algebras A(E, q) with dim E = 3
7.6. Clifford algebras A(E, q) with dim E = 4
7.7. Clifford algebras A(E, q) with dim E = 5
7.7. Clifford algebras A(E, q) with dim E = 5
7.8. algebras A6, B7, A7 and A8
8. Spin
8.1. Clifford groups
8.2. Pin and Spin groups
8.3. Replacing q by
q
8.4. spin group for odd dimensions
8.5. Spin groups, for d = 2
8.6. Spin groups, for d = 3
8.7. Spin groups, for d = 4
8.8. group Spin5
8.9. Examples of spin groups for d ≥ 6
8.10. Table of results
pt. Three Some Applications
9. Some applications to physics
9.1. Particles with spin 1/2
9.2. Dirac operator
9.3. Maxwell's equations
9.4. Dirac equation
10. Clifford analyticity
10.1. Clifford analyticity
10.2. Cauchy's integral formula
10.3. Poisson kernels and the Dirichlet problem
10.4. Hilbert transform
10.5. Augmented Dirac operators
10.6. Subharmonicity properties
10.7. Riesz transform
10.8. Dirac operator on a Riemannian manifold
11. Representations of Spind and SO(d)
11.1. Compact Lie groups and their representations
11.2. Representations of SU (2)
11.3. Representations of Spind and SO(d) for d ≤ 4
12. Some suggestions for further reading.