240934 An Introduction to Lie groups and Lie algebras (BS) (SoSe 2012)

An introduction to Lie groups and Lie algebras

Lie group is a group which is also a smooth manifold such that the group operations are smooth maps. A typical example is the group SLn (R) of n by n real matrices of determinant one. Lie groups play a central role in mathematics with strong links to Number theory, Ergodic theory, Differential equations, Physics and more. The aim of this course is to give an introduction of the basic theory Linear Lie groups.

Syllabus

Linear Lie groups, one-parameter subgroups, the exponential map, Lie algebras and their connections to Lie groups, structure theorems for Lie algebras, Killing form, representations of Lie algebras, root systems, classification of semisimple complex Lie algebras, and forms (if time permits).

Bibliography

• B. Hall, Lie groups, Lie algebras and representations: An elementary introduction (Springer).
• D. Bump, Lie groups (Springer).
• A. Onishchik and E. B. Vinberg, Lie groups and algebraic groups (Springer).
• J. P. Serre, Complex semisimple Lie algebras (Springer).
• J. Humphreys, Introduction to Lie Algebras and representation theory.