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Clifford Algebras: an Introduction

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ISBN-10: 1107422191

ISBN-13: 9781107422193

Edition: 2011

Authors: D. J. H. Garling

List price: $65.95
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Book details

List price: $65.95
Copyright year: 2011
Publisher: Cambridge University Press
Publication date: 6/23/2011
Binding: Paperback
Pages: 210
Size: 6.00" wide x 9.00" long x 0.50" tall
Weight: 0.880
Language: English

D. J. H. Garling is a Fellow of St John's College and Emeritus Reader in Mathematical Analysis at the University of Cambridge, in the Department of Pure Mathematics and Mathematical Statistics.

Introduction
The algebraic environment
Groups and vector spaces
Groups
Vector spaces
Duality of vector spaces
Algebras, representations and modules
Algebras
Group representations
The quaternions
Representations and modules
Module homomorphisms
Simple modules
Semi-simple modules
Multilinear algebra
Multilinear mappings
Tensor products
The trace
Alternating mappings and the exterior algebra
The symmetric tensor algebra
Tensor products of algebras
Tensor products of super-algebras
Quadratic forms and Clifford algebras
Quadratic forms
Real quadratic forms
Orthogonality
Diagonalization
Adjoint mappings
Isotropy
Isometries and the orthogonal group
The case d = 2
The Cartan-Dieudonn� theorem
The groups SO(3) and SO(4)
Complex quadratic forms
Complex inner-product spaces
Clifford algebras
Clifford algebras
Existence
Three involutions
Centralizers, and the centre
Simplicity
The trace and quadratic form on A(E, q)
The group G(E, q) of invertible elements of A(E, q)
Classifying Clifford algebras
Frobenius' theorem
Clifford algebras A(E, q) with dim E = 2
Clifford's theorem
Classifying even Clifford algebras
Cartan's periodicity law
Classifying complex Clifford algebras
Representing Clifford algebras
Spinors
The Clifford algebras A<sub>k</sub>, k
The algebras B<sub>k,k+i</sub> and A<sub>k, k+1</sub>
The algebras A<sub>k+1,k</sub> and A<sub>k +2,k</sub>
Clifford algebras A(E, q) with dim E = 3
Clifford algebras A(E, q) with dim E = 4
Clifford algebras A(E, q) with dim E = 5
The algebras A<sub>6</sub>, B<sub>7</sub>, A<sub>7</sub> and A<sub>8</sub>
Spin
Clifford groups
Pin and Spin groups
Replacing q by �q
The spin group for odd dimensions
Spin groups, for d = 2
Spin groups, for d = 3
Spin groups, for d = 4
The group Spin<sub>5</sub>
Examples of spin groups for d &#8805; 6
Table of results
Some Applications
Some applications to physics
Particles with spin 1/2
The Dirac operator
Maxwell's equations
The Dirac equation
Clifford analyticity
Clifford analyticity
Cauchy's integral formula
Poisson kernels and the Dirichlet problem
The Hilbert transform
Augmented Dirac operators
Subharmonicity properties
The Riesz transform
The Dirac operator on a Riemannian manifold
Representations of Spin<sub>d</sub> and SO(d)
Compact Lie groups and their representations
Representations of SU(2)
Representations of Spind and SO(d) for d &#8804; 4
Some suggestions for further reading
References
Glossary
Index