18.755 | Fall 2004 | Graduate

Introduction to Lie Groups


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

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).

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

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

Course Info

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