O.K.Sheinman

Basic Representation Theory

Group symmetry is one of the most fundamental geometrical properties of the real world. It suffices to say that three of the four fundamental types of physical interactions (namely, electromagnetic, weak, and strong) are controlled by symmetries of the first three special unitary groups: U(1), SU(2), and SU(3), respectively. The representation theory of these groups and of some related structures will be considered in the proposed course.

Syllabus

1. Representations of finite groups:

• Schur lemma
• complete reducibility
• 1-st and 2-d Burnside theorems
• characters and equivalence

2. Representations of the special unitary group:

• Peter-Weyl theorem
• Weyl formula for characters
• representations of SU(2)
• the hydrogen atom
• quantum numbers and the periodic system of elements

3. Representations of the Lie algebra sl(2,C):

• finite-dimensional irreducible modules
• Casimir operator

• Representations of the simplest Kac-Moody algebra $\widehat{sl}^(2,{\bf C})$ and combinatorial analysis.

Books:

1. W. Fulton, J. Harris. Representation Theory. A first course, Springer-Verlag, 1991