Students will learn the basic structure theory of simple Lie algebras over the complex numbers, as well as the theory of their highest weight representations. This will develop students' understanding of these fundamental objects in algebra, and give them some hands-on experience computing representations and proving some powerful (and quite beautiful!) results.
Students will learn the basic structures of simple Lie algebras over the complex numbers, including classification, root systems, Cartan subalgebras, Cartan/triangular decomposition, Dynkin diagrams, Weyl groups, and the Killing form. We will develop a highest weight theory of representations, including Verma modules and enveloping algebras. We will end with Weyl's character formula for finite-dimensional simple modules.
This is an alternating years course, taught first in AY2020, and subsequently in AY2022, etc.
1) Definition and key examples of Lie algebras
2) Structure theory of Lie algebras
3) Root systems, Dynkin diagrams, and Weyl groups
4) Classification of finite-dimensional (semi-)simple Lie algebras
5) Highest weight modules, simple modules, and Verma modules
6) Weyl's character formula for finite-dimensional simple modules
A solid grasp of undergraduate linear algebra, as well as experience following long proofs and constructing your own proofs. Students must be very comfortable with proofs in order to understand the material in this course and complete the homework questions adequately. If you are unsure, please discuss this further with your academic mentor. Some prior knowledge of the representation theory of finite groups will also be helpful when grappling with analogous results for Lie algebras, but it is not completely necessary.