SUBJECT

Lie groups and symmetric spaces

lecture + practical

master

Faculty of Science

Mathematics MSc

6+3

Semesters 1-4

Autumn/Spring semester

Lie groups and their Lie algebras. Exponential mapping, adjoint representation, Hausdorff-Baker-Campbell formula. Structure of Lie algebras; nilpotent, solvable, semisimple, and reductive Lie algebras. Cartan subalgebras, classification of semisimple Lie algebras.

Differentiable structure on a coset space. Homogeneous Riemannian spaces. Connected compact Lie groups as symmetric spaces. Lie group formed by isometries of a Riemannian symmetric space. Riemannian symmetric spaces as coset spaces. Constructions from symmetric triples. The exact description of the exponential mapping and the curvature tensor. Totally geodesic submanifolds and Lie triple systems. Rank of a symmetric space. Classification of semisimple Riemannian symmetric spaces. Irreducible symmetric spaces.

• S. Helgason: Differential geometry, Lie groups, and symmetric spaces. Academic Press, New York, 1978.

Further reading:

• O. Loos: Symmetric spaces I–II. Benjamin, New York, 1969.