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 | 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, R ) | 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 |