MATH 6490: Lie Algebras Spring 2013

Texts: none

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 an introductory 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. We will highlight the relation between Lie groups and Lie algebras throughout the course.

Basic structure and properties of Lie algebras; theorems of Lie and Engel.
Nilpotent solvable and reductive Lie algebras.
The relation between Lie groups and Lie algebras; algebraic groups
Enveloping algebras and differential operators
The structure of semisimple algebras
Representation theory of semisimple Lie algebras; Lie algebra cohomology
Compact semisimple groups and their representation theory. Chevalley groups
Quantum groups, Kac-Moody algebras and their representations theory

References

N. Bourbaki, Groupes et algebres de Lie, Hermann, Paris, 1971
V. Chari and A. Pressley, A guide to quantum groups
J. Dixmier, Enveloping algebras
S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978.
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
J-P. Serre, Complex semisimple Lie algebras