| Topology and Differential Geometry | p. 1 |
| Introduction to Part I | p. 3 |
| Topology | p. 5 |
| Preliminaries | p. 5 |
| Topological Spaces | p. 6 |
| Metric spaces | p. 9 |
| Basis for a topology | p. 11 |
| Closure | p. 12 |
| Connected and Compact Spaces | p. 13 |
| Continuous Functions | p. 15 |
| Homeomorphisms | p. 17 |
| Separability | p. 18 |
| Homotopy | p. 21 |
| Loops and Homotopies | p. 21 |
| The Fundamental Group | p. 25 |
| Homotopy Type and Contractibility | p. 28 |
| Higher Homotopy Groups | p. 34 |
| Differentiable Manifolds I | p. 41 |
| The Definition of a Manifold | p. 41 |
| Differentiation of Functions | p. 47 |
| Orientability | p. 48 |
| Calculus on Manifolds: Vector and Tensor Fields | p. 50 |
| Calculus on Manifolds: Differential Forms | p. 55 |
| Properties of Differential Forms | p. 59 |
| More About Vectors and Forms | p. 62 |
| Differentiable Manifolds II | p. 65 |
| Riemannian Geometry | p. 65 |
| Frames | p. 67 |
| Connections, Curvature and Torsion | p. 69 |
| The Volume Form | p. 74 |
| Isometry | p. 74 |
| Integration of Differential Forms | p. 77 |
| Stokes' Theorem | p. 80 |
| The Laplacian on Forms | p. 83 |
| Homology and Cohomology | p. 87 |
| Simplicial Homology | p. 87 |
| De Rham Cohomology | p. 100 |
| Harmonic Forms and de Rham Cohomology | p. 103 |
| Fibre Bundles | p. 105 |
| The Concept of a Fibre Bundle | p. 105 |
| Tangent and Cotangent Bundles | p. 111 |
| Vector Bundles and Principal Bundles | p. 112 |
| Bibliography for Part I | p. 117 |
| Group Theory and Structure and Representations of Compact Simple Lie Groups and Algebras | p. 119 |
| Introduction to Part II | p. 121 |
| Review of Groups and Related Structures | p. 123 |
| Definition of a Group | p. 123 |
| Conjugate Elements, Equivalence Classes | p. 124 |
| Subgroups and Cosets | p. 24 |
| Invariant (Normal) Subgroups, the Factor Group | p. 125 |
| Abelian Groups, Commutator Subgroup | p. 126 |
| Solvable, Nilpotent, Semisimple and Simple Groups | p. 127 |
| Relationships Among Groups | p. 129 |
| Ways to Combine Groups - Direct and Semidirect Products | p. 131 |
| Topological Groups, Lie Groups, Compact Lie Groups | p. 132 |
| Review of Group Representations | p. 135 |
| Definition of a Representation | p. 135 |
| Invariant Subspaces, Reducibility, Decomposability | p. 136 |
| Equivalence of Representations, Schur's Lemma | p. 138 |
| Unitary and Orthogonal Representations | p. 139 |
| Contragredient, Adjoint and Complex Conjugate Representations | p. 140 |
| Direct Products of Group Representations | p. 144 |
| Lie Groups and Lie Algebras | p. 147 |
| Local Coordinates in a Lie Group | p. 147 |
| Analysis of Associativity | p. 148 |
| One-parameter Subgroups and Canonical Coordinates | p. 151 |
| Integrability Conditions and Structure Constants | p. 155 |
| Definition of a (real) Lie Algebra: Lie Algebra of a given Lie Group | p. 157 |
| Local Reconstruction of Lie Group from Lie Algebra | p. 158 |
| Comments on the G → G Relationship | p. 160 |
| Various Kinds of and Operations with Lie Algebras | p. 161 |
| Linear Representations of Lie Algebras | p. 165 |
| Complexification and Classification of Lie Algebras | p. 171 |
| Complexification of a Real Lie Algebra | p. 171 |
| Solvability, Levi's Theorem, and Cartan's Analysis of Complex (Semi) Simple Lie Algebras | p. 173 |
| The Real Compact Simple Lie Algebras | p. 180 |
| Geometry of Roots for Compact Simple Lie Algebras | p. 183 |
| Positive Roots, Simple Roots, Dynkin Diagrams | p. 189 |
| Positive Roots | p. 189 |
| Simple Roots and their Properties | p. 189 |
| Dynkin Diagrams | p. 194 |
| Lie Algebras and Dynkin Diagrams for SO(2l), SO(2l+1), USp(2l), SU(l + 1) | p. 197 |
| The SO(2l) Family - Dl of Cartan | p. 197 |
| The SO(2l + 1) Family - Bl of Cartan | p. 201 |
| The USp(2l) Family - Cl of Cartan | p. 203 |
| The SU(l + 1) Family - Al of Cartan | p. 207 |
| Coincidences for low Dimensions and Connectedness | p. 212 |
| Complete Classification of All CSLA Simple Root Systems | p. 215 |
| Series of Lemmas | p. 216 |
| The allowed Graphs | p. 220 |
| The Exceptional Groups | p. 224 |
| Representations of Compact Simple Lie Algebras | p. 227 |
| Weights and Multiplicities | p. 227 |
| Actions of E()#x003C;/sub> and SU(2) - the Weyl Group | p. 228 |
| Dominant Weights, Highest Weight of a UIR's | p. 230 |
| Fundamental UIR's, Survey of all UIR's | p. 233 |
| Fundamental UIR's for Al, Bl, Cl, Dl | p. 234 |
| The Elementary UIR's | p. 240 |
| Structure of States within a UIR | p. 241 |
| Spinor Representations for Real Orthogonal Groups | p. 245 |
| The Dirac Algebra in Even Dimensions | p. 246 |
| Generators, Weights and Reducibility of U(S) - the spinor UIR's of Dl | p. 248 |
| Conjugation Properties of Spinor UIR's of Dl | p. 250 |
| Remarks on Antisymmetric Tensors Under Dl = SO(2l) | p. 252 |
| The Spinor UIR's of Bl = SO(2l + 1) | p. 257 |
| Antisymmetric Tensors under Bl = SO(2l + 1) | p. 260 |
| Spinor Representations for Real Pseudo Orthogonal Groups | p. 261 |
| Definition of SO(q,p) and Notational Matters | p. 261 |
| Spinor Representations S() of SO(p, q)for p + q = 2l | p. 262 |
| Representations Related to S() | p. 264 |
| Behaviour of the Irreducible Spinor Representations S±() | p. 265 |
| Spinor Representations of SO(p, q) for p + q = 2l + 1 | p. 266 |
| Dirac, Weyl and Majorana Spinors for SO(p, q) | p. 267 |
| Bibliography for Part II | p. 273 |
| Index | p. 275 |
| Table of Contents provided by Ingram. All Rights Reserved. |
