18.755 | Fall 2004 | Graduate

Introduction to Lie Groups


1 Historical Background and Informal Introduction to Lie Theory
2 Differentiable Manifolds, Differentiable Functions, Vector Fields, Tangent Spaces
3 Tangent Spaces; Mappings and Coordinate Representation


4 Affine Connections

Parallelism; Geodesics

Covariant Derivative

5 Normal Coordinates

Exponential Mapping

6 Definition of Lie Groups

Left-invariant Vector Fields

Lie Algebras

Universal Enveloping Algebra

7 Left-invariant Affine Connections

The Exponential Mapping

Taylor’s Formula in a Lie Group Formulation

The Group GL (n, R)

8 Further Analysis of the Universal Enveloping Algebra

Explicit Construction of a Lie Group (locally) from its Lie Algebra

Exponentials and Brackets

9 Lie Subgroups and Lie Subalgebras

Closer Subgroups

10 Lie Algebras of some Classical Groups

Closed Subgroups and Topological Lie Subgroups

11 Lie Transformation Groups

A Proof of Lie’s Theorem

12 Homogeneous Spaces as Manifolds

The Adjoint Group and the Adjoint Representation

13 Examples

Homomorphisms and their Kernels and Ranges

14 Examples

Non-Euclidean Geometry

The Associated Lie Groups of Su (1, 1) and Interpretation of the Corresponding Coset Spaces

15 The Killing Form

Semisimple Lie Groups

16 Compact Semisimple Lie Groups

Weyl’s Theorem proved using Riemannian Geometry

17 The Universal Covering Group
18 Semi-direct Products

The Automorphism Group as a Lie Group

19 Solvable Lie Algebras

The Levi Decomposition

Global Construction of a Lie Group with a given Lie Algebra

20 Differential 1-forms

The Tensor Algebra and the Exterior Algebra

21 Exterior Differentiation and Effect of Mappings

Cartan’s Proof of Lie’s Third Theorem

22 Integration of Forms

Haar Measure and Invariant Integration on Homogeneous Spaces

23 Maurer-Cartan Forms

The Haar Measure in Canonical Coordinates

24 Invariant Forms and Harmonic Forms

Hodge’s Theorem

25 Real Forms

Compact Real Forms, Construction and Significance

26 The Classical Groups and the Classification of Simple Lie Algebras, Real and Complex

Course Info

As Taught In
Fall 2004
Learning Resource Types
notes Lecture Notes
assignment_turned_in Problem Sets with Solutions