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. Cover Table of Contents 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 Homomorphism 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-Dieudonné Theorem The Groups SO(3) and SO(4) Complex Quadratic Forms Complex Inner-product Spaces Clifford Algebras Cilfford Algebras Existence Three Involutions Centralizers, and the Center 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 𝒜ₖ,ₖ The Algebras ℬₖ,ₖ₊₁ and 𝒜ₖ,ₖ₊₁ The Algebras 𝒜ₖ₊₁,ₖ and 𝒜ₖ₊₂,ₖ 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 𝒜₆, ℬ₇, 𝒜₇ and 𝒜₈ 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 Spin₅ Examples of Spin Groups for d ≥ 6 Table of Results Some Applications Some Applications to Physics Particles with Spin 1/2 The Dirac Operator Maxwell's Equations 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_d and SO(d) Compact Lie Groups and their Representations Representations of SU(2) Representations of Spin_d and SO(d) for d ≥ 4 Some Suggestions for Further Reading References Glossary Index