MATH 649 — Fall 2000
Lie Algebras

Lie groups, Lie algebras and their representations play an important role in much of mathematics, particularly in number theory, mathematical physics and topology.

This is a basic course in Lie algebras. The prerequisites are a basic knowledge of algebra and linear algebra at the honors undergraduate level. The first five topics are standard. I hope to be able to devote substantial time to the last one. Ken Brown will teach a course on Kac-Moody algebras in the spring.

• Basic structure and properties of Lie algebras; theorems of Lie and Engel.
• Nilpotent solvable and reductive Lie algebras.
• Enveloping Algebras and Differential Operators
• The structure of semisimple algebras
• Representation theory of semisimple Lie algebras; Lie algebra cohomology
• Quantum groups, Kac-Moody algebras and their representations theory. algebra cohomology.

References

J. Dixmier, Enveloping algebras

N. Jacobson, Lie algebras

J. Humphreys, Introduction to Lie algebras and representation theory

S. Helgason, Differential geometry, Lie groups and symmetric spaces

V. Kac, Infinite dimensional Lie algebras

G. Lusztig, Introduction to quantum groups

J-P. Serre, Complex semisimple Lie algebras

Last modified: April 7, 2003