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