Department of Mathematics

Lie Groups III

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Prerequisites: Lie Groups I, Lie Groups II (Symmetric Spaces).

Contents: Symmetric spaces of non-compact type: Roots and root systems. Characterizations of the Weyl group and its action on the Weyl chambers. Geometric boundary and its characterization via Busemann functions. The space SL(n,R)/SO(n) and the Imbedding Theorem. Geometric Iwasawa decomposition. Tits metric on the geometric boundary.

Organizers: Prof. Dr. Marc Burger, Stephan Tornier.


The seminar takes place on Thursdays, 1 p.m. to 4 p.m. at HG D3.2, until 13.11.2014.

Date Speaker Contents
18.09.2014 Stephan Tornier Introduction, discussion of possible topics and distribution of talks.

25.09.2014 Matteo Felder Root systems of symmetric spaces of non-compact type. Equivalence of several definitions of the Weyl group.
02.10.2014 Stephan Tornier Abstract root systems. The Weyl group acts simply transitively on the set of Weyl chambers.

09.10.2014 Pascal Wild Geometric boundary of a symmetric space of non-compact type. Equivalent characterization via Busemann functions.
16.10.2014 Leyli Mammadova Explicit description of the example SL(n,R)/SO(n). A sketch of the Imbedding Theorem.

23.10.2014 Giuliano Basso Parabolic subgroups: Preparations for the Iwasawa decomposition.

30.10.2014 Andreas Wieser Proof of the Iwasawa decomposition. Illustrations in the case SL(n,R)/SO(n).

06.11.2014 Raphael Schumacher Iwasawa decomposition on the level of Lie algebras. Angular metric and Tits metric on the geometric boundary.

13.11.2014 Stephan Tornier Towards the Tits Building, abstract buildings.


Werner Ballmann: Lectures on spaces of nonpositive curvature.
Armand Borel: Semisimple groups and Riemannian symmetric spaces.
Martin Bridson and André Haefliger: Metric spaces of non-positive curvature.
Patrick B. Eberlein: Geometry of nonpositively curved manifolds.
Patrick B. Eberlein: Structure of manifolds of nonpositive curvature.
Sigurdur Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces.


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