# Assignments

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

Main Assignments (from this and previous years)

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 |