Lie Groups

10 credits

Syllabus, Master's level, 1MA048

This course has been discontinued.

A revised version of the syllabus is available.

Code: 1MA048
Education cycle: Second cycle
Main field(s) of study and in-depth level: Mathematics A1F
Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
Finalised by: The Faculty Board of Science and Technology, 15 March 2007
Responsible department: Department of Mathematics

Entry requirements

BSc, Algebraic Structures, Linear Algebra II, Topology

Learning outcomes

In order to pass the course (grade 3) 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–CampbellHausdorff'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 and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.