Syllabus for Lie Groups

Learning outcomes

On completion of the course, the student should be able to:

• give an account of the following concepts: differentiable manifold, vector field and Lie bracket;
• explain the concepts of topological group and Lie group and give examples of such groups;
• use the Baker-Campbell-Hausdorff formula;
• describe the Lie algebra of a given Lie group;
• translate properties of the Lie algebra to properties of the associated Lie group;
• define and exemplify the following Lie group concepts: nilpotent, solvable, and semisimple;
• explain the relation between, on the one hand, general Lie groups and, on the other hand, nilpotent, solvable and semisimple Lie groups.

Content

Differentiable manifolds and Lie groups, especially closed subgroups of the real and the complex

general group. Classical families of simply connected compact groups. Vector fields and the Lie algebra of a Lie group, the exponential map, Baker-Campbell-Hausdorff's formula. The relation between a Lie group and its corresponding Lie algebra. Nilpotent, solvable and semisimple Lie groups. Representations of Lie groups.

Instruction

Lectures and problem solving sessions.

Assessment

Written examination at the end of the course and assignments given during the course.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.