Lie Algebras
Students wanting to learn graduate-level algebra, and especially representation theory. A solid grasp of linear algebra will be assumed, as well as an ability to understand and construct quite sophisticated mathematical proofs.
By the end of this course, students will be able to:
Define and illustrate key structures in Lie algebra theory, including root systems, Cartan subalgebras, and Dynkin diagrams.
Construct and analyze representations of Lie algebras, with a focus on highest weight modules, Verma modules, and enveloping algebras.
Apply the theory of finite-dimensional simple Lie algebras to compute and interpret Weyl’s character formula.
Demonstrate proficiency in constructing rigorous mathematical proofs related to the structure and representation theory of Lie algebras.
Connect abstract algebraic concepts to concrete computational techniques, gaining hands-on experience with representation-theoretic calculations.
Learn the fundamental objects in algebra, especially representation theory, with hands-on experience in computing representations and constructing sophisticated proofs for some powerful (and quite beautiful!) results. Practice focuses on the basic structures of simple Lie algebras over the complex numbers, as well as the theory of highest weight representation including Verma modules and enveloping algebras, concluding with Weyl's character formula for finite-dimensional simple modules. Additional topics include root systems, Cartan subalgebras, Cartan/triangular decomposition, Dynkin diagrams, and the Killing form.
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
Homework: 100%. There will be roughly 6 sets of homework problems during the term.
Solid undergraduate linear algebra. Confident in following and constructing proofs. Some prior knowledge of the representation theory of finite groups is helpful but not completely necessary. Discuss this carefully with your academic mentor.
Introduction to Lie Algebras and Representation Theory, by James Humphreys
Representation Theory: A first course, by William Fulton and Joe Harris
Introduction to Lie algebras, by Karin Erdmann and Mark Wildon
not available AY2025