Lie Algebra (Math 7360)

Schedule

Section	Place	Time		Final Exam Schedule
Math 7360-140	Parker 248	MWF	14:00-14:50	Fri	May 6	16:00-18:30

Instructor: Dr. Huajun Huang

• Office:  207 Parker Hall
• Phone:  (344) 844-5974
• Email: huanghu (at) auburn
• Office Hours:  MWF 11:00 am-12:00 pm.

Information

• Textbook: Introduction to Lie Algebras and Representation Theory, by James E. Humphreys.
• Reference Books:

1. Complex semisimple Lie algebras, Jean-Pierre Serre, Springer, Berlin Heidelberg, 1987.

2. Lie Algebras, Nathan Jacobson, Dover, New York, 1979.

3. Lie groups beyond an introduction, Chapters I & II, Anthony W. Knapp, 2002.

4. Modular Lie Algebras and Their Representations, Helmut Strade and Rolf Farnsteiner, Pure and Applied Mathematics, Marcel Dekker, New York, 1988.

• Prerequisites:  A good comprehension of linear algebra and abstract algebra is required.
• Description: This course aims to give a general introduction of Lie algebras and the structure of semisimple Lie algebras. The course contents include:

· Lie algebras, subalgebras, linear Lie algebras and linear groups, adjoint representations, abelian Lie algebras.

· Ideals, centers, derived algebras, simple algebras, quotient algebras, homomorphisms, representations.

· Solvable Lie algebras, radical, semisimple Lie algebras, nilpotent Lie algebras, Engel's Theorem.

· Lie's Theorem, Jordan-Chevalley decomposition, Cartan's Criterion for solvable Lie algebras.

· Killing forms, radicals, simple ideals, inner derivations, abstract Jordan decompositions.

· Lie algebra modules, module homomorphisms, irreducible modules, completely reducible modules, Schur's Lemma, Casimir element, Weyl's Theorem, preservation of Jordan decomposition.

· Representations of sl(2,F).

· Toral subalgebras, root space decomposition (Cartan decomposition), properties of maximal toral subalgebras.

· Reflections, root system, roots, dual root system, base, Weyl chambers, Weyl group.

· Cartan matrix, Cartan integers, Coxeter graphs, Dynkin diagrams, classification of simple Lie algebras, automorphisms of root systems.

· Weights, root lattices, half sum of positive roots, highest weight, saturated sets of weights.

• Grade:  Overall Score = homeworks (50%) + classroom performance (10%) + final project (40%).

Grade	A	B	C	D	F
Score	90-100	80-89	70-79	60-69	0-59

Materials

• Lecture Notes:

§ 1.1, 1.2, 1.3, 1.4,

§ 2.1, 2.2, 2.3, 2.4,

§ 3.1, 3.2, 3.3.

• Homework Assignments:

§ Chapter 1 Chapter 2 Chapter 3

§ Final Project