دانلود کتاب 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 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"--Back cover. Read more... 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