Dr. Christian Lange
Wintersemester 2020/21
Lecture
Thursday 12pm-2pm, online
Exercise session
Wednesday 2pm-4pm, online
Description
A Lie group is a group which is also a smooth manifold, in way that the group multiplication and the inversion are smooth maps. Examples of Lie groups are matrix groups like the linear group GL(n,R) or the orthogonal group O(n). Lie groups play an important role in many parts of mathematics and physics. They almost inevitably occur when a problem is invariant under symmetries. In the lecture we will discuss structural results about Lie groups as well as examples and applications. An important tool will be the Lie functor, which assigns to every Lie group its so-called Lie algebra. The Lie algebra is an infinitesimal version of the Lie group that encodes surprisingly much information about it, and that can be well studied via methods from linear algebra. This will allow us to establish a classification of complex semi-simple Lie algebras and of compact Lie groups. In doing so we will follow an approach suggested by Cartan that is more geometric than the standard one.
More specifically, we plan to cover the following topics
- Lie correspondence between Lie groups and -algebras.
- Exponential map and adjoint representation.
- Structure of (solvable-, nilponent- and) compact Lie groups.
- Geometry of Lie groups with bi-invariant metrics: Geodesics, curvature and Weyl's theorem.
- Maximal tori.
- Complex semisimple Lie algebras, existence and conjugacy of compact real forms.
- Cartan subalgebras and root space decomposition.
- Cartan-Killing classification.
Prerequisites
The course assumes knowledge of basic group theory, topology I and smooth manifolds. In particular, students should be familiar with notions like tangent bundles, vector fields, etc. A more precise description together with a source well suited for a recapulation of these concepts can be found in the moodle course. Basic knowledge of Riemannian geometry is advantageous, but it can also be acquired during the lecture or some details can be taken for granted.
Moodle
Further information and access to the course material can be found in moodle; see lsf for the login details.