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

Information

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.

      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

A

B

C

D

F

Score

90-100

80-89

70-79

60-69

0-59

Materials

  1.1, 1.2, 1.3, 1.4,

  2.1, 2.2, 2.3, 2.4,

  3.1, 3.2, 3.3.

  Chapter 1 Chapter 2 Chapter 3

  Final Project