Mathematics 405 - Lie Algebras

Lie groups and Lie algebras appear naturally in the study of symmetries of geometric objects. Lie algebras carry local information and can be studied using tools from linear algebra. Finite dimensional Lie groups can be constructed using techniques from calculus and group theory.

This class serves as a first introduction to Lie groups and Lie algebras . We will examine the structure of finite dimensional matrix Lie groups, the exponential and differentiation maps, as well as compact Lie groups. We will also study Lie algebras, ideals and homomorphisms, nilpotent and solvable Lie algebras, Cartan subalgebras, semisimplicity and 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. 4 class meetings per week.

Requisite: MATH 350 or consent of the instructor. Limited to 18 students. Spring semester. Professor Contreras Palacios.