Section 12: existence of Thm 12.5; principal fiber bundles
Section 13: proper and free actions
Section 14: coset spaces
skipped Section 15: orbits of smooth actions.
Section 16: the Baire property: background reading
Section 17: normal subgroups, the isomorphism theorem

1. Lec 1A - Introduction to Lie Groups
2. Lec 1B - Lie Groups Definitions and Basic Properties
3. Lec 2A - Invariant Vector Fields & The Exponential Map
4. Lec 2B - The Lie Algebra of a Lie Group
5. Lec 3A - The Lie Algebra of a Lie Group II
6. Lec 3B - Commuting Elements & Component of the Identity
7. Lec 4A - Commutative Lie Groups
8. Lec 4B - Lie Subgroups & Analytic Subgroup Theorem
9. Lec 5A - Closed Subgroups
10. Lec 5B - The Groups SU(2) and SO(3)
11. Lec 6A - Group Actions and Orbit Spaces
12. Lec 6B - Actions of Principal Fiber Bundle Type & Smoothness Principle
13. Lec 7A - Proper and Free Actions & Coset Spaces
14. Lec 7B - Baire Property, Normal Subgroups, and Isomorphism Theorem
15. Lec 8A - Densities and Integration
16. Lec 8B - Densities and Integration II
17. Lec 9A - left Haar measure, bi-invariant Haar measure
18. Lec 9B - Finite Dimensional Continuous Representations
19. Lec 10A - Schur's Lemma & Orthogonality
20. Lec 10B - Formulation of the Peter-Weyl theorem
21. Lec 11A - Characters and Multiplicities
22. Lec 11B - Class Functions and Classical Fourier Series
23. Lec 12A - Lie Algebra Representations
24. Lec 12B - Irreducible Representations of sl(2,C)
25. Lec 13A - Highest Weight of Irreducible Representation
26. Lec 13B - Compact Lie Algebras
27. Lec 14A - Automorphisms and Derivations
28. Lec 14B - Characterization of Compact Semisimple Lie Algebras by Killing form
29. Lec 15A - Reflections and the Weyl group
30. Lec 15B - Classification of Irreducible Representations
31. Lec 16A - Cartan Integers
32. Lec 16B - Dynkin Diagrams & Classification of Root Systems

In this course we will begin by studying the basic properties of Lie groups. Much of the structure of a connected Lie group is captured by its Lie algebra, which may be defined as the algebra of left invariant vector fields. The exponential map will be introduced, and the relation between the structure of a Lie group and its Lie algebra will be investigated. Actions of Lie groups will be studied.

After this introduction we will focus on compact Lie groups and the integration theory on them. The groups SU(2) and SO(3) will be discussed as basic examples. We will study representation theory and its role in the harmonic analysis on a Lie group. The classification of the irreducible representations of SU(2) will be studied.

The final part of the course will be devoted to the classification of compact Lie algebras. Key words are: root system, finite reflection group, Cartan matrix, Dynkin diagram. The course will be concluded with the formulation of the classification of irreducible representations by their highest weight and with the formulation of Weyl's character formula.

A Lie group is a group with the additional structure of a differentiable manifold for which the group operation is differentiable. The name Lie group comes from the Norwegian mathematician M. Sophus Lie (1842-1899) who was the first to study these groups systematically in the context of symmetries of partial differential equations.

The theory of Lie groups has developed vastly in the course of the previous century. It plays a vital role in the description of symmetries in Physics (quantum physics, elementary particles), Geometry and Topology, and Number Theory (automorphic forms).