Clifford Algebras : an Introduction.
A straightforward introduction to Clifford algebras, providing the necessary background material and many applications in mathematics and physics.
Saved in:
| Online Access: | Electronic book from EBSCO |
|---|---|
| Main Author: | |
| Format: | eBook |
| Language: | English |
| Published: | Cambridge : Cambridge University Press, 2011. |
| Series: | London Mathematical Society Student Texts, 78.
|
| Subjects: |
Table of Contents:
- Cover; London Mathematical Society Student Texts 78: Clifford Algebras: An Introduction; Title; Copyright; Contents; Introduction; PART ONE: 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 The 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 The trace.
- 3.4 Alternating mappings and the exterior algebra3.5 The symmetric tensor algebra; 3.6 Tensor products of algebras; 3.7 Tensor products of super-algebras; PART 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; 4.8 The Cartan-Dieudonné theorem; 4.9 The 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 centre5.5 Simplicity; 5.6 The trace and quadratic form on A(E, q); 5.7 The group G(E; q) of invertible elements of A(E, q); 6: Classifying Clifford algebras; 6.1 Frobenius' theorem; 6.2 Clifford algebras A(E, q) with dimE = 2; 6.3 Clifford's theorem; 6.4 Classifying even Clifford algebras; 6.5 Cartan's periodicity law; 6.6 Classifying complex Clifford algebras; 7: Representing Clifford algebras; 7.1 Spinors; 7.2 The Clifford algebras Ak, k; 7.3 The algebras Bk, k+1 and Ak, k+1; 7.4 The 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 = 47.7 Clifford algebras A(E, q) with dim E = 5; 7.8 The 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 The 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 The group Spin5; 8.9 Examples of spin groups for d>= 6; 8.10 Table of results; PART THREE: SOME APPLICATIONS; 9: Some applications to physics; 9.1 Particles with spin 1/2; 9.2 The Dirac operator; 9.3 Maxwell's equations; 9.4 The Dirac equation.
- 10: Clifford analyticity10.1 Clifford analyticity; 10.2 Cauchy's integral formula; 10.3 Poisson kernels and the Dirichlet problem; 10.4 The Hilbert transform; 10.5 Augmented Dirac operators; 10.6 Subharmonicity properties; 10.7 The Riesz transform; 10.8 The 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; The algebraic environment; Quadratic spaces; Clifford algebras.