## Math 427: Lie groups

### Basic Information

**Instructor: **Eugene Lerman
**e-mail:**lerman@math.uiuc.edu
**Homepage:**
`https://faculty.math.illinois.edu/~lerman`

** Course page:**
`https://faculty.math.illinois.edu/~lerman/427/427syl.html`

**Office:** 334 Illini Hall
**Phone:** 244-9510
**Class meets:** MWF 9 am
in 155 Altgeld Hall
**Office hours:** MW 11-12 and by appointment

## Prerequisites

Mathematics 423 or consent of instructor.
If you have any questions or concerns, please contact me by e-mail.

## Course outline

This course is an introduction to Lie theory. The first part of the
course will cover the foundations:

- Examples of Lie groups and their Lie algebras
- Homomorphisms
- Lie subgroups
- Covering spaces
- Closed subgroups
- Continuous homomorphisms
- The exponential map
- Representations; the adjoint and coadjoint representations
- Homogeneous manifolds
- Group actions, orbits, adjoint and coadjoint orbits

The remainder of the course will touch on the following topics:
Abelian, nilpotent, solvable and semi-simple Lie groups, Levi
decomposition, semi-direct products. Compact Lie groups, Peter-Weyl
theorem, maximal tori, Weyl groups. Ado's theorem: every finite
dimensional Lie algebra is a matrix Lie algebra. Cartan's theorem
(Lie's third theorem): every finite dimensional Lie algebra is a Lie
algebra of a Lie group.

## Texts

The official text is Lie groups beyond an
introduction by A. W. Knapp

There is also a recommended text:
- Lie groups, Lie algebras, and their
representations by V. S. Varadarajan

## Grades

The course grade will be based on homework. Homework will be assigned
weekly. One problem per homework will be graded.

Last modified: Tue Aug 24 13:51:30 CDT