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Clifford algebras : an introduction

Author: D J H Garling
Publisher: Cambridge ; New York : Cambridge University Press, 2011.
Series: London Mathematical Society student texts, 78.
Edition/Format:   Book : EnglishView all editions and formats
Database:WorldCat
Summary:
"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  Read more...
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Document Type: Book
All Authors / Contributors: D J H Garling
ISBN: 9781107422193 9781107096387 1107096383 1107422191
OCLC Number: 707967489
Description: vii, 200 pages : illustrations ; 23 cm.
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 & ge; 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 & le; 4 --
12. Some suggestions for further reading.
Series Title: London Mathematical Society student texts, 78.
Responsibility: D.J.H. Garling.

Abstract:

A straightforward introduction to Clifford algebras, providing the necessary background material and many applications in mathematics and physics.  Read more...

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'... it became clear that Garling has spotted a need for a particular type of book, and has delivered it extremely well. Of all the books written on the subject, Garling's is by some way the most Read more...

 
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schema:description"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 & ge; 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 & le; 4 -- 12. Some suggestions for further reading."@en
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