# Assignments

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).

Assignments by Class Session
SES # TOPICS PROBLEMS
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
3

Tangent Spaces; Mappings and Coordinate Representation

Submanifolds

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