Textbook: Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics #9), by James E. Humphreys

Additional references: Introduction to Lie Algebras, by Karin Erdmann and Mark Wildon; Lie Algebras of Finite and Affine Type, by Roger Carter; Lie Groups and Lie Algebras Chapters 4-6, by Nicolas Bourbaki; Lie Algebras, by Nathan Jacobson

Prerequisites: Mastery of linear algebra and of the methods of abstract algebra. In particular, students should have a good knowledge of notions related to eigenvalues, bilinear forms, euclidean spaces, canonical forms, tensor products, and a good knowledge of the basics of module theory. Much of the prerequisite theory from linear algebra can be found in Appendix A of Erdmann and Wildon's book.

Course overview: This course in an introduction to the structure and representation theory of semisimple Lie algebras over the complex numbers. Besides appearing throughout mathematics and physics (e.g., in the study of Lie groups, differentiable manifolds, algebraic groups, quantum groups, finite groups of Lie type, particle physics), the theory of Lie algebras, and complex semisimple Lie algebras in particular, is quite beautiful. In fact, the classification theorem for complex semisimple Lie algebras served as a model for the eventual classification of all finite simple groups, one of the most important mathematical results of the twentieth century!

I anticipate that we will cover parts of Chapters I-III and V-VI of the textbook. In particular, we will study

• Definitions and basic concepts related to Lie algebras
• Nilpotent and solvable Lie algebras (theorems of Engel, Lie, and Cartan)
  
• Structure of semisimple Lie algebras (Killing form, Weyl's theorem, weight space decomposition)
• Root systems (axiomatics, the Weyl group, classification of root systems, weights)
• Universal enveloping algebras (PBW theorem)
  
• Classification of semisimple Lie algebras over the complex numbers
• Finite-dimensional representation theory of complex semisimple Lie algebras
  

My goal is for us to get as far as possible in discussing the representation theory of complex semisimple Lie algebras, and hopefully to get as far as discussing Weyl's character formula. To get this far in the time available, we may move fairly rapidly through some parts of the material. It will occasionally be up to you to fill in gaps from the lectures by reading the relevant portions of the textbook on your own.

Grading: Grades will be assigned based on attendance and class participation (including homework).

Course schedule and homework: Homework will be assigned and collected on a regular basis.

• August 15: §1 Definitions and first examples. Homework: §1 # 6, 9 (do only one of A, B, C, or D), 11, 12
  
• August 17: §2 Ideals and homomorphisms. Homework: §2 # 1, 2, 4, 5, 10 (turn in homework for §1-2 on August 24)
• August 19: Finish §2, begin §3 Solvable and nilpotent Lie algebras.
• August 22: §3 Solvable and nilpotent Lie algebras.
  
• August 24: Finish §3 Solvable and nilpotent Lie algebras. Homework: §3 # 4, 6, 7, 10 (updated)
  
• August 26: §4 Theorems of Lie and Cartan. Homework: §4 # 1, 3, 4
• August 29: §4 Theorems of Lie and Cartan. Turn in homework for §3-4 on September 2 or September 7.
• August 31: §4 Theorems of Lie and Cartan, §5 Killing form.
• September 7: §6.1 Modules. Homework §5 # 7 (along the way, compute the matrix of the Killing form for sl(3,F)). §6 # 3.
• September 9: Student presentations.
• September 12: §6.1 Modules. (For the homework, use §6 # 7 when doing §5 # 7.)
• September 14: §6.2-6.3 Casimir element and Weyl's theorem. Homework: §6 # 1 (just the part for sl3).
• September 16: §6.3 Weyl's theorem. HW3: §5 # 7, §6 # 1 (just the part for sl3), 3, 6, 7. Turn in HW3 on September 23.
• September 19: §6.3 Weyl's theorem, §7 Representations of sl2(F), §7.1 Weights and maximal vectors.
• September 21: §7.2 Classification of irreducible modules. HW4: §7 # 2, 5, 6, 7. Turn in HW4 on September 30?
• September 23: §8.1 Root space decomposition, toral subalgebras
  
• September 26: §8.2-8.3 Orthogonality properties for roots
  
• September 28: §8.3 Othogonality properties for roots
  
• September 30: §8.4-8.5, Integrality and rationality properties for roots.
• October 3: §8.5 Rationality properties for roots, §9 Root systems. HW5: §8 # 1, 2, 9, 10
• October 5: §9 Root systems
• October 7: §10.1 Simple roots. Turn in HW5 by October 17.
• October 10: §10 Simple roots and the Weyl group.
• October 12: §10 The Weyl group and irreducible root systems. HW6: §9 # 3, 4. §10 # 2, 9. §11 # 3.
• October 14: §11-12 Classification of irreducible root systems.
• October 17: §14.1 Isomorphism theorem, reduction to the simple case. Target deadline for HW6 is October 31.
  
• October 19: §14.2 Isomorphism theorem, the simple case.
  
• October 21: §14.2 Isomorphism theorem, the simple case.
• October 24: Finish §14.
  
• Ocotber 26: §17 The universal enveloping algebra.
• October 28: No class (Fall break). Target deadline for HW6 is October 31.
• October 31: §17 PBW theorem and free Lie algebras
• November 2: §18 Generators and relations.
  
• November 4: §18 Serre's theorem.
• November 7: §18 Serre's theorem.
  
• November 9: §18 Serre's theorem. HW7: §14 # 2, 5, §17 # 4, §18 # 1, 4. Target deadline for HW7 is November 28.
  
• November 11: Finish §18, begin discussion of highest weight representations for semisimple Lie algebras.