# Lie Groups and Lie Algebra

### Course Description

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).

Prof. Erik van den Ban in the first lecture of the course.
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### Video Lectures & Study Materials

Visit the official course website for more study materials: http://www.staff.science.uu.nl/~ban00101/lie2012/lie2012.html

# Lecture Play Lecture
1 Lec 1A - Introduction to Lie Groups (46:34) Play Video
2 Lec 1B - Lie Groups Definitions and Basic Properties (45:38) Play Video
3 Lec 2A - Invariant Vector Fields & The Exponential Map (46:34) Play Video
4 Lec 2B - The Lie Algebra of a Lie Group (51:24) Play Video
5 Lec 3A - The Lie Algebra of a Lie Group II (42:44) Play Video
6 Lec 3B - Commuting Elements & Component of the Identity (50:34) Play Video
7 Lec 4A - Commutative Lie Groups (45:31) Play Video
8 Lec 4B - Lie Subgroups & Analytic Subgroup Theorem (50:29) Play Video
9 Lec 5A - Closed Subgroups (50:45) Play Video
10 Lec 5B - The Groups SU(2) and SO(3) (46:50) Play Video
11 Lec 6A - Group Actions and Orbit Spaces (49:39) Play Video
12 Lec 6B - Actions of Principal Fiber Bundle Type & Smoothness Principle (44:57) Play Video
13 Lec 7A - Proper and Free Actions & Coset Spaces (57:06) Play Video
14 Lec 7B - Baire Property, Normal Subgroups, and Isomorphism Theorem (41:27) Play Video
15 Lec 8A - Densities and Integration (39:15) Play Video
16 Lec 8B - Densities and Integration II (52:44) Play Video
17 Lec 9A - left Haar measure, bi-invariant Haar measure (48:50) Play Video
18 Lec 9B - Finite Dimensional Continuous Representations (48:51) Play Video
19 Lec 10A - Schur's Lemma & Orthogonality (50:30) Play Video
20 Lec 10B - Formulation of the Peter-Weyl theorem (41:55) Play Video
21 Lec 11A - Characters and Multiplicities (55:26) Play Video
22 Lec 11B - Class Functions and Classical Fourier Series (41:08) Play Video
23 Lec 12A - Lie Algebra Representations (50:55) Play Video
24 Lec 12B - Irreducible Representations of sl(2,C) (44:11) Play Video
25 Lec 13A - Highest Weight of Irreducible Representation (48:41) Play Video
26 Lec 13B - Compact Lie Algebras (46:43) Play Video
27 Lec 14A - Automorphisms and Derivations (45:35) Play Video
28 Lec 14B - Characterization of Compact Semisimple Lie Algebras by Killing form (47:59) Play Video
29 Lec 15A - Reflections and the Weyl group (51:31) Play Video
30 Lec 15B - Classification of Irreducible Representations (40:50) Play Video
31 Lec 16A - Cartan Integers (50:05) Play Video
32 Lec 16B - Dynkin Diagrams & Classification of Root Systems (45:35) Play Video