- Course begins: April
17th 24th, 2014! (Note: There course will not meet the first week first two weeks!)
- Course time: 8:00-10:00 Thursday + Friday
- Course location: M102
- Exercise session: 16:00-18:00 Friday
- Exercise session location: M101
- Office hours by appointment in M206
- There will be an oral exam at the end of the course.
This will be a first course on Lie theory, i.e., Lie groups and Lie algebras, and the only prerequisites will be some experience with algebra and differential geometry (groups and smooth manifolds).
In general the study of infinite groups is extremely difficult, however if we ask that the group has the additional structure of a manifold (a topological space that locally looks like \(\mathbb{R}^n\)) then the problem becomes more manageable. In this case we say we have a Lie group. Such groups naturally occur in physics as symmetries. For example the circle is the group of rotations of a plane.
With some further restrictions one can actually classify such groups by restricting to a neighborhood of the identity and examining the induced structure on a vector space approximation. The structure on this vector space is a called a Lie algebra and the classification of compact simply connected Lie groups comes from a classification of a corresponding restricted class of Lie algebras. One of the goals of this course is to understand this classification result.
For many applications one would like to understand how such groups can act on vector spaces (e.g., the circle acting on the plane). Such actions are called representations and this theory is well understood in the case of compact simply connected Lie groups and semisimple Lie algebras. The other main goal of this course is to understand these actions.