Contents -- Part I: Topology and Differential Geometry -- Introduction to Part I -- 1 Topology -- 1.1 Preliminaries -- 1.2 Topological Spaces -- 1.3 Metric spaces -- 1.4 Basis for a topology -- 1.5 Closure -- 1.6 Connected and Compact Spaces. -- 1.7 Continuous Functions -- 1.8 Homeomorphisms. -- 1.9 Separability -- 2 Homotopy -- 2.1 Loops and Homotopies -- 2.2 The Fundamental Group -- 2.3 Homotopy Type and Contractibility -- 2.4 Higher Homotopy Groups -- 3 Differentiable Manifolds I -- 3.1 The Definition of a Manifold -- 3.2 Differentiation of Functions -- 3.3 Orientability -- 3.4 Calculus on Manifolds: Vector and Tensor Fields -- 3.5 Calculus on Manifolds: Differential Forms -- 3.6 Properties of Differential Forms. -- 3.7 More About Vectors and Forms -- 4 Differentiable Manifolds II -- 4.1 Riemannian Geometry -- 4.2 Frames -- 4.3 Connections, Curvature and Torsion -- 4.4 The Volume Form -- 4.5 Isometry -- 4.6 Integration of Differential Forms -- 4.7 Stokes'Theorem -- 4.8 The Laplacian on Forms -- 5 Homology and Cohomology -- 5.1 Simplicial Homology -- 5.2 De Rham Cohomology -- 5.3 Harmonic Forms and de Rham Cohomology -- 6 Fibre Bundles -- 6.1 The Concept of a Fibre Bundle -- 6.2 Tangent and Cotangent Bundles -- 6.3 Vector Bundles and Principal Bundles -- Bibliography for Part I -- Part II: Group Theory and Structure and Representations of Compact Simple Lie Groups and Algebras -- Introduction to Part II -- 7 Review of Groups and Related Structures -- 7.1 Definition of a Group -- 7.2 Conjugate Elements, Equivalence Classes -- 7.3 Subgroups and Cosets -- 7.4 Invariant (Normal) Subgroups, the Factor Group -- 7.5 Abelian Groups, Commutator Subgroup -- 7.6 Solvable, Nilpotent, Semi simple and Simple Groups. -- 7.7 Relationships Among Groups -- 7.8 Ways to Combine Groups - Direct and Semidirect Products.

7.9 Topological Groups, Lie Groups, Compact Lie Groups -- Exercises for Chapter 7 -- 8 Review of Group Representations -- 8.1 Definition of a Representation -- 8.2 Invariant Subspaces, Reducibility, Decomposability -- 8.3 Equivalence of Representations, Schur's Lemma -- 8.4 Unitary and Orthogonal Representations. -- 8.5 Contragredient, Adjoint and Complex Conjugate Representations -- 8.6 Direct Products of Group Representations -- Exercises for Chapter 8 -- 9 Lie Groups and Lie Algebras -- 9.1 Local Coordinates in a Lie Group. -- 9.2 Analysis of Associativity -- 9.3 One-parameter Subgroups and Canonical Coordinates -- 9.4 Integrability Conditions and Structure Constants -- 9.5 Definition of a (real) Lie Algebra: Lie Algebra of a given Lie Group -- 9.6 Local Reconstruction of Lie Group from Lie Algebra -- 9.7 Comments on the G G Relationship -- 9.8 Various Kinds of and Operations with Lie Algebras. -- Exercises for Chapter 9 -- 10 Linear Representations of Lie Algebras -- Exercises for Chapter 10 -- 11 Complexification and Classification of Lie Algebras -- 11.1 Complexification of a Real Lie Algebra -- 11.2 Solvability, Levi's Theorem, and Cartan's Analysis of Complex (Semi) Simple Lie Algebras -- Levi Splitting Theorem -- Theorem (Cartan, 1894) -- 11.3 The Real Compact Simple Lie Algebras -- Exercises for Chapter 11 -- 12 Geometry of Roots for Compact Simple Lie Algebras -- 13 Positive Roots, Simple Roots, Dynkin Diagrams -- 13.1 Positive Roots -- 13.2 Simple Roots and their Properties -- 13.3 Dynkin Diagrams. -- Exercises for Chapters 12 and 13 -- 14 Lie Algebras and Dynkin Diagrams for SO(2l), SO(2l+1), USp(2l), SU(l + 1) -- 14.1 The SO(2l) Family - Dl of Cartan -- 14.2 The SO(2l + 1) Family - Bl of Cartan -- 14.3 The USp(2l) Family - Gl of Cartan -- 14.4 The SU(l + 1) Family - Al of Cartan.

14.5 Coincidences for low Dimensions and Connectedness -- Exercises for Chapter 14 -- 15 Complete Classification of All CSLA Simple Root Systems -- 15.1 Series of Lemmas -- 15.2 The allowed Graphs r -- 15.3 The Exceptional Groups -- 16 Representations of Compact Simple Lie Algebras -- 16.1 Weights and Multiplicities -- 16.2 Actions of En and SU(2)(a) - the Weyl Group -- 16.3 Dominant Weights, Highest Weight of a UlR -- 16.4 Fundamental UIR's, Survey of all UIR's -- 16.5 Fundamental UIR's for AI, Bl , Gl, Dl -- 16.6 The Elementary UIR's -- 16.7 Structure of States within a UIR -- Exercises for Chapter 16 -- 17 Spinor Representations for Real Orthogonal Groups -- 17.1 The Dirac Algebra in Even Dimensions. -- 17.2 Generators, Weights and Reducibility of U(S) - the spinor UIR's of Dl -- 17.3 Conjugation Properties of Spinor UIR's of Dl -- 17.4 Remarks on Antisymmetric Tensors Under Dl = SO(2l) -- 17.5 The Spinor UIR's of Bl = SO(2l + 1) -- 17.6 Antisymmetric Tensors under Bl = SO(2l + 1) -- 18 Spinor Representations for Real Pseudo Orthogonal Groups -- 18.1 Definition of SO(q,p) and Notational Matters. -- 18.2 Spinor Representations S(A) of SO(p, q) for p + q = 2l -- 18.3 Representations Related to S(A) -- 18.4 Behaviour of the Irreducible Spinor Representations S±(A) -- 18.5 Spinor Representations of SO(p, q) for p + q = 2l + 1 -- 18.6 Dirac, Weyl and Majorana Spinors for SO(p, q) -- Exercises for Chapters 17 and 18 -- Bibliography for Part II -- Index.