241460 Seminar on Matrix Groups - An Introduction to Lie Theory (S) (WiSe 2019/2020)

The goal of the course is to introduce the participants to the basic ideas and applications of the theory of Lie groups. The focus on matrix Lie groups allows for many of the more technical aspects to be circumvented, and opens the course to geometers, analysts, algebraists, and even physicists and chemists. Of course matrix Lie groups have important applications in all of these areas, and anyone pursuing study in any them will soon find deeper use for the material covered on this course.
A course outline begins with the basic definitions of real and complex matrix groups and a study of their basic topological features. Next we associate with each matrix group its Lie algebra, and come to understand the relationship between the algebraic structure of this object and the topology of the original matrix group. Of course this brings us to the matrix exponential and the analytic side of the course.

At this point it would be nice to come to understand some concrete examples and applications, and depending on the audience it might be worthwhile to study the rotation and Lorentz groups, or make a diversion to consider Clifford algebras and spinor groups.

Finally we will end the course by studying the aspects of the theory that are more special to it. We will define maximal tori and end the course by discussing the classification of the compact matrix groups.

Requirements for participation, required level

The formal requirements for the course are fairly minimal. Mainly we require that students should be acquainted with basic linear algebra and calculus. Knowledge of some elementary group theory and some further real/complex analysis would be helpful, although is not required.

Bibliography

References for this course include the book 'Matrix Groups - An Introduction to Lie Group Theory' by Andrew Baker, and 'Matrix Groups for Undergraduates' by Kristopher Tapp. Both are available in the Bielefeld University library.