Math 8710 Lie Algebra 2024 Fall
General Information
Information | |
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Instructor | Kang Lu (wbt4qn) at Kerchof Hall 229 |
Office Hours | Mondays 3:30-4:45PM and Tuesdays 4-5PM and by appointment |
Syllabus | Syllabus |
Textbooks
- J. Humphreys, Introduction to Lie Algebras and Representation Theory, GTM 9, Springer.
- R. Carter, Lie Algebras of Finite and Affine Type, Cambridge University Press.
Prerequisites:
Advanced linear algebra (Math 4651 level) and basic familiarity with notions of groups, rings and modules.
Schedules
- 08/28/2024, Definition and examples of Lie algebras
- 09/02/2024, Engel’s theorem
- 09/04/2024, Lie’s theorem
- 09/09/2024, Jordan-Chevalley decomposition and Cartan’s criterion
- 09/11/2024, Killing form, semisimple Lie algebras, and complete reducibility
- 09/16/2024, Representations of $\mathfrak{sl}_2$ and classical Lie algebras
- 09/18/2024, Cartan subalgebra and root space decomposition
- 09/23/2024, Properties of roots
- 09/25/2024, Root system and Weyl groups
- 09/30/2024, Simple roots, Weyl groups
- 10/02/2024, Cartan matrix and Dynkin diagram
- 10/09/2024, Classification of Cartan matrices
- 10/16/2024, Root sysmtems of classical Lie algebras
- 10/21/2024, The existence and uniqueness theorems
- 10/23/2024, Universal enveloping algebras and PBW theorem
- 10/28/2024, Verma modules and their properties
- 10/30/2024, Classification of finite-dimensional irreducible representations I
- 11/04/2024, Classification of finite-dimensional irreducible representations II + Casimir elements
- 11/06/2024, Central characters
- 11/11/2024, Weyl character formulas
- 11/13/2024, Representations of Lie algebras of type A I
- 11/18/2024, Representations of Lie algebras of type A II
- 11/20/2024, Representations of Lie algebras of type BCD
- 11/25/2024, Presentations 1
- 12/02/2024, Presentations 2
- 12/04/2024, Presentations 3