240157 Lie Algebras and Lie Groups (V) (SoSe 2017)

Lie groups are groups with a differentiable manifold structure such that the group operations are compatible with the manifold structure. As key tools in the description of continuous symmetries they are ubiquitous in mathematics, but also play a role in mathematical physics.

One may associate to a Lie group a Lie algebra -- explicitly constructible as the group's tangent space at the identity element --, which captures much of the "local" structure. "Exponential" and "logarithm" maps translate between Lie groups and their associated Lie algebra, facilitating a sort of transfer of several problems about Lie groups in the (linear) realm of Lie algebras.

One of the main objectives of the course will be the classification of (complex, say) simple Lie algebras in terms of Dynkin diagrams, one of the most pervasive results of modern algebra.

Bibliography

1. J. Tits, Liesche Gruppen und Algebren, Springer-Verlag, 1983
2. J. Hilgert, K.-H. Neeb, Structure and geometry of Lie groups, Springer Monographs in Mathematics, 2012
3. K. Erdman, M. J. Wildon, Introduction to Lie Algebras, Springer, 2006