# Math 6324 Lie algebras

#### Description

In a Lie group, in contrast to a finite group, the elements depend on

continuously varying parameters. Lie groups are used to describe

symmetries that are not discrete, like reflections, but vary

continuously, like rotations.

While the elements in a Lie group usually do not depend linearly on the

parameters, it turns out that almost the entire group structure can be

captured in a linear object, the so-called Lie algebra associated to the

Lie group. The essential properties of Lie algebras can be encoded in a

few simple axioms, which makes it possible to study them abstractly.

Over the complex numbers, it is possible to classify the so-called

simple Lie algebras completely. This remarkable classification was

carried out by Wilhelm Killing at the end of the nineteenth century and

had a large impact on many fields in modern mathematics. The course will

give an introduction to Lie algebras and this classification.

#### Objectives

The objective of the course is to introduce the student to the

foundations of the algebraic theory of Lie algebras. Basic topics

covered at the beginning are solvability, nilpotency, simplicity, and

semisimplicity. Building on these notions, the course will usually

progress to cover the classification of finite-dimensional simple Lie

algebras over algebraically closed fields of characteristic zero

mentioned above, and the concepts that are associated with it, such as

Cartan subalgebras, the Killing form, root systems, and Dynkin diagrams.

#### Prerequisites

Advanced Linear Algebra (6351) or equivalent.

#### Textbooks

- N. Jacobson:
*Lie algebras*, Dover, Mineola, 1979 - J. E. Humphreys:
*Introduction to Lie algebras and representation theory*,

Grad. Texts Math., Vol. 9, Springer, Berlin, 1972