Clifford Algebras: An Introduction ( London Mathematical Society Student Texts )

Publication series ：London Mathematical Society Student Texts

Author： D. J. H. Garling  

Publisher： Cambridge University Press‎

Publication year： 2011

E-ISBN: 9781139066389

P-ISBN(Paperback): 9781107096387

Subject： O151.24 vector algebra, the factor algebra, algebraic invariant theory

Keyword： 数学

Language： ENG

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Clifford Algebras: An Introduction

Description

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.

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London Mathematical Society Student Texts 78: Clifford Algebras: An Introduction

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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 algebra

3.5 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 centre

5.5 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 = 4

7.7 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 analyticity

10.1 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

Clifford algebras and harmonic analysis

Clifford analysis on Riemannian manifolds

Spin groups and representation theory

Applications to physics

References

Glossary

Index

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