The course grade is based 100% on the homework assignments.

### Main Assignments (from this and previous years)

A2, A7, A8, C2, D1, E1 in chapter I.

A1, A2, A3, A6 (i)-(iii), B1, C2, C5, D3 in chapter II.

### Solutions to Assignments

Solutions for sessions 2-5, 14, and 20-22 may be found in Chapter I Solutions (PDF). Solutions for problems for sessions 6-13, 15, 16, and 23-25 may be found in Chapter II Solutions (PDF).

SES # | TOPICS | PROBLEMS |
---|---|---|

1 | Historical Background and Informal Introduction to Lie Theory | Read the first two papers listed under Additional Readings |

2 | Differentiable Manifolds, Differentiable Functions, Vector Fields, Tangent Spaces | Suggested Problems: A2, 3, 8 |

3 |
Tangent Spaces; Mappings and Coordinate Representation Submanifolds |
Suggested Problems: A4, A5, A7, D3 |

4 |
Affine Connections
Parallelism; Geodesics Covariant Derivative |
Suggested Problems: C2, D2 |

5 |
Normal Coordinates
Exponential Mapping |
Suggested Problem: C5 |

6 |
Definition of Lie groups
Left-invariant Vector Fields Lie Algebras Universal Enveloping Algebra |
Suggested Problems: A1, A2, A3 |

7 |
Left-invariant Affine Connections
The Exponential Mapping Taylor’s Formula in a Lie Group Formulation The Group GL (n, |
Suggested Problems: A6 (i), (ii), (iii), B1 |

8 |
Further Analysis of the Universal Enveloping Algebra
Explicit Construction of a Lie Group (locally) from its Lie Algebra Exponentials and Brackets |
Suggested Problems: B4, B5 |

9 |
Lie Subgroups and Lie Subalgebras
Closer Subgroups |
Suggested Problems: C2, C4 |

10 |
Lie Algebras of some Classical Groups
Closed Subgroups and Topological Lie Subgroups |
Suggested Problems: C1, D1 |

11 |
Lie Transformation Groups
A Proof of Lie’s Theorem |
Suggested Problems: C5, C6 |

12 |
Homogeneous Spaces as Manifolds
The Adjoint Group and the Adjoint Representation |
Suggested Problems: D3 (i)-(iv) |

13 | Examples Homomorphisms and their Kernels and Ranges | Suggested Problems: A4, C3 |

14 |
Examples Non-Euclidean Geometry
The Associated Lie Groups of Su (1, 1) and Interpretation of the Corresponding Coset Spaces |
Suggested Problem: E1 |

15 |
The Killing Form
Semisimple Lie Groups |
Suggested Problem: D2 |

16 |
Compact Semisimple Lie Groups
Weyl’s Theorem proved using Riemannian Geometry |
Suggested Problem: B3 |

17 | The Universal Covering Group | No Problems Assigned |

18 |
Semi-direct Products
The Automorphism Group as a Lie Group |
No Problems Assigned |

19 |
Solvable Lie Algebras
The Levi Decomposition Global Construction of a Lie Group with a given Lie Algebra |
No Problems Assigned |

20 |
Differential 1-Forms
The Tensor Algebra and the Exterior Algebra |
Suggested Problems: B1, B2, B3 |

21 |
Exterior Differential and Effect of Mappings
Cartan’s Proof of Lie Third Theorem |
Suggested Problems: B4, B5, B6 |

22 |
Maurer-Cartan Forms
The Haar Measure in Canonical Coordinates |
Suggested Problem: C4 |

23 |
Maurer-Cartan Forms
The Haar Measure in Canonical Coordinates |
Suggested Problems: E1, E3, F1, F2, F3 |

24 |
Invariant Forms and Harmonic Forms
Hodge’s Theorem |
Suggested Problems: E2, F4, F5, F6 |

25 |
Real Forms
Compact Real Forms, Construction and Significance |
Suggested Problems: G1, G3 |

26 | The Classical Groups and the Classification of Simple Lie Algebras, Real and Complex | Read the third paper listed under Additional Readings |