Mathematics 405 - Lie Algebras

Lie algebras originally arose as a way of studying certain continuous transformation groups called Lie groups. Lie algebras are simpler objects than Lie groups since they can be studied using tools from linear algebra, yet they still provide a lot of information about their associated Lie groups. This class serves as a first introduction to the theory of Lie algebras. We will examine the structure of finite dimensional Lie algebras, ideals and homomorphisms, nilpotent and solvable Lie algebras, Cartan subalgebras, semisimplicity, root systems, and the classification of semisimple Lie algebras. This classification is not only a fundamental result in Lie Theory, but is also an archetype of classifications that appear in other areas of math. More amazingly, this classification is embodied in simple combinatorial pictures called Dynkin diagrams, which underlie surprisingly disparate fields, such as geometric group theory, quiver representation theory, and string theory.

Requisite: MATH 350 or consent of the instructor. Limited to 18 students. Fall semester. Visiting Professor Naqvi.