Math 8710 Lie Algebra 2024 Fall

General Information

Information
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

  1. J. Humphreys, Introduction to Lie Algebras and Representation Theory, GTM 9, Springer.
  2. 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