Lie Algebra (Math 7360)

## Schedule

 Section Place Time 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:

§

§