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 |