**Lie-Gruppen/Lie groups WS2015/16**

**Information**

Mastervorlesung (oder fortgeschritten Bachelorvorlesung) über Lie-Gruppen. Sprache: Deutsch oder Englisch (nach Bedarf). Alle schriftliche Kommunikationen werden auf english sein!

A Lie group is a group with the additional structure of a differentiable manifold and for which the group operation is differentiable. These objects have been studied systematically by Sophus Lie (1842 - 1899) to understand the symmetries of partial differential equations. Nowadays Lie groups arise in many branches of mathematics, such as differential geometry, PDEs, mathematical physics, number theory, harmonic analysis, special function theory, ... Compact Lie groups are closely related to linear algebraic groups, which are in turn important in algebraic geometry. For example, the finite dimensional representations of a compact Lie group and its complexification (a linear algebraic group) are more or less the same.

The **aim** of this course is to provide students with a basic knowledge of Lie theory. A lot of information of a Lie group is already contained in its Lie algebra. The link between these objects is given by the exponential map. Another way to understand a Lie group is via its representations. To this end, we will study the representation theory of SU(2). Finally, we will study the classification of compact Lie groups via their root systems.

**Outlook**: After this basic course we intend to offer follow-up course(s) that are more specialized (e.g. representation theory, structure theory, invariant theory, ...). There are also many interesting thesis subjects in these directions (BA/MA).

**Organization**

- The
**lectures**are on Thursdays, 9:15 - 10:45 in Q1.213. - The
**exercise classes**are on Wednesdays, 13:00 - 14:00, in D1.320. Job Kuit will be around to answer your questions. First meeting: 21.10.2015, last meeting: 10.2.2015.**Here**is a collection of exercises (courtesy of Erik van den Ban). - We will mostly follow the
**lecture notes**by Erik van den Ban. - The
**prerequisites**are: basic knowledge of analysis, topology, differentiable manifolds and group theory.**Here**are some notes (courtesy of Erik van den Ban) that cover most of the material that we will be needing from manifold theory. - The
**examination**consists of 3 hand-in exercises (40%) and an oral exam (60%). **Credits**after passing the examination: 5ECTs.**Material discussed in the lectures**.

**Literature**

- Duistermaat, J. J.; Kolk, J. A. C., Lie groups. Universitext. Springer-Verlag, Berlin, 2000.
- Bröcker, T; tom Dieck, T., Representations of compact Lie groups. Graduate Texts in Mathematics, 98. Springer-Verlag, New York, 1985.
- Humphreys, J, Introduction to Lie algebras and representation theory. Springer, 1972.