| LEC # | TOPICS |
|---|---|
| 1 | Historical Background and Informal Introduction to Lie Theory |
| 2 | Differentiable Manifolds, Differentiable Functions, Vector Fields, Tangent Spaces |
| 3 | Tangent Spaces; Mappings and Coordinate Representation Submanifolds |
| 4 | Affine Connections Parallelism; Geodesics Covariant Derivative |
| 5 | Normal Coordinates Exponential Mapping |
| 6 | Definition of Lie Groups Left-invariant Vector Fields Lie Algebras Universal Enveloping Algebra |
| 7 | Left-invariant Affine Connections The Exponential Mapping Taylor's Formula in a Lie Group Formulation The Group GL (n, R) |
| 8 | Further Analysis of the Universal Enveloping Algebra Explicit Construction of a Lie Group (locally) from its Lie Algebra Exponentials and Brackets |
| 9 | Lie Subgroups and Lie Subalgebras Closer Subgroups |
| 10 | Lie Algebras of some Classical Groups Closed Subgroups and Topological Lie Subgroups |
| 11 | Lie Transformation Groups A Proof of Lie's Theorem |
| 12 | Homogeneous Spaces as Manifolds The Adjoint Group and the Adjoint Representation |
| 13 | Examples Homomorphisms and their Kernels and Ranges |
| 14 | Examples Non-Euclidean Geometry The Associated Lie Groups of Su (1, 1) and Interpretation of the Corresponding Coset Spaces |
| 15 | The Killing Form Semisimple Lie Groups |
| 16 | Compact Semisimple Lie Groups Weyl's Theorem proved using Riemannian Geometry |
| 17 | The Universal Covering Group |
| 18 | Semi-direct Products The Automorphism Group as a Lie Group |
| 19 | Solvable Lie Algebras The Levi Decomposition Global Construction of a Lie Group with a given Lie Algebra |
| 20 | Differential 1-forms The Tensor Algebra and the Exterior Algebra |
| 21 | Exterior Differentiation and Effect of Mappings Cartan's Proof of Lie's Third Theorem |
| 22 | Integration of Forms Haar Measure and Invariant Integration on Homogeneous Spaces |
| 23 | Maurer-Cartan Forms The Haar Measure in Canonical Coordinates |
| 24 | Invariant Forms and Harmonic Forms Hodge's Theorem |
| 25 | Real Forms Compact Real Forms, Construction and Significance |
| 26 | The Classical Groups and the Classification of Simple Lie Algebras, Real and Complex |