Lie Group and Lie Algebra

邱雁南老师的MAT7073。每周四晚连上三小时，上完就可以困晕过去了（）

I Lie algebra

1 Introduction

2 Properties of solvable and nilpotent Lie algebra

• 2.1 Existence of (generalized) weight space
• 2.2 Engel’s theorem

3 The Killing form - describe solvable and semi-simple

4 Classification of semi-simple Lie algebra

• 4.1 Cartan subalgebra
• 4.2 The structure of finite dimensional complex semi-simple Lie subalgebra
• 4.3 Root system
• 4.4 Weyl chambers
• 4.5 Dynkin diagram

5 Representation theory

• 5.1 Finite dimensional representations of sl(2, C)
• 5.2 Representation theory of semi-simple Lie algebra

II Lie group

6 Lie group

• 6.1 Basic setup for Lie group
• 6.2 Relate Lie algebra with Lie group
• 6.3 Basic theorems in the Lie theory
• 6.4 Haar measure
• 6.5 Lie algebra - Lie group correspondence

Appendix

• 0.0 Exercise
• 0.1 An introduction to manifolds
• 0.2 Differential
• Carter - 2005 - Lie algebras of finite and affine type.pdf