MAT 1120HF Lie groups and Lie algebras

Classes: MWF 1-2 BA 6183

The theory of Lie groups and Lie algebras is a classical and well-established subject of mathematics. The plan for this course is to give an introduction to the foundations of this theory, with emphasis on compact Lie groups and semi-simple Lie algebras. The course may serve as a preparation towards J. Kamnitzer's more advanced course MAT1196HS. Prerequisites: Manifolds, algebraic topology.

Topics to be discussed will (hopefully) include:
• the classical Lie groups
• actions of Lie groups and Lie algebras
• the Lie algebra of a Lie group
• exponential map
• enveloping algebras, PBW theorem
• semi-simple Lie algebras, relation with compact Lie groups
• maximal torus, Cartan subalgebras
• representations of sl(2,C)
• roots and weights
• Weyl group, chambers
• Coxeter-Dynkin diagrams, classification
• Finite-dimensional epresentations of semi-simple Lie algebras
• Example: Representations of sl(n,C)
• Weyl character formula

Problem set #1 Due: October 6, 2010. Solutions #1 (Brad Hannigan-Daley)

Problem set #2 Due: November 3, 2010.

Problem set #3 (revised, 11/25) Due: December 10, 2010.

Our main references are:

1. A. Kirillov Jr.: An Introduction to Lie Groups and Lie Algebras , Cambridge University Press

2. T. Broecker. T. tom Dieck: Representations of compact Lie groups , Springer

1. D. Bump: Lie Groups, Springer

2. J. J. Duistermaat, J. Kolk: Lie Groups, Springer

3. V. V. Gorbatsevich, A. L. Onishchik, E. B. Vinberg: Foundations of Lie Theorey and Lie Transformation groups , Springer

4. A. Knapp: Lie Groups Beyond an Introduction , Birkhauser

5. Y. Kosmann-Schwarzbach: Groups and Symmetries, Springer

6. S. Sternberg, Lie algebras , available from his website.