Course Content

The main textbook for the class is:

• Compact Lie Groups by M.R. Sepanski.

Students are strongly encouraged but not required to own a copy of this book, as we will follow its content rather closely. All problem sets will be posted below, and many exercise will be copied from the textbook (rather than, say, simply a list of exercises from the textbook).

Homework assignments

• Assignment 1
• Assignment 2
• Assignment 3
• Assignment 4
• Assignment 5
• Assignment 6. Note: Question 3 is mildly wrong as stated, or at least ambiguous. There is a bijection between symmetric compact and symmetric affine dynkin diagrams, which is of the form "add/subtract a node". But the choice of where to add/subtract the node is not arbitrary.

Approximate agenda

1. Compact Lie groups: basic notions and examples.
2. Representations: irreducibility, unitarizability.
3. Harmonic analys: Schur orthogonality and characters; the Peter-Weyl theorem; Fourier theory.
4. Lie algebras: definitions and constructions.
5. Abelian subgroups.
6. Roots, lattices, and the Weyl group.
7. Highest weight theory and the Weyl character formula.