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.

Impressum | Webmaster | Letzte Änderungen am : 21.04.2016