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, |

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 |

## Calendar

## Course Info

##### Learning Resource Types

*notes*Lecture Notes

*assignment_turned_in*Problem Sets with Solutions