Introductory description

The concept of continuous symmetry suggested by Sophus Lie had an enormous influence on many branches of mathematics and physics in the twentieth century. Created first as a tool in a small number of areas (e.g. PDEs) it developed into a separate theory which influences many areas of modern mathematics such as geometry, algebra, analysis, mechanics and the theory of elementary particles, to name a few.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Topological Groups
• Manifolds
• Lie Groups
• Lie Algebras
• Tangent Space
• The Lie Algebra of a Lie Group
• Integrating Vector Fields and the Exponential Map
• Lie Subgroups
• Continuity implies Smoothness
• Outlook: Differential Geometry and Representation Theory

Learning outcomes

By the end of the module, students should be able to:

• After taking this module, students will have working knowledge of Lie groups. They will be familiar with connections of Lie groups to Algebra, Differential Geometry and Representation Theory. Students will learn examples of Lie groups and their Lie algebras.

Subject specific skills

Students who successfully complete the module will have developed understanding of the Lie group theory and their connections with the Lie algebra theory. This will include:

• familiarity with the basic concepts of manifolds, vector fields and topological groups to the extend they are used in the theory of Lie groups;
• detailed knowledge of all classical simple Lie groups and their Lie algebras;
• understanding of the invariant vector fields on a Lie group, the exponential map, the Cartan theorem and the orbit-stabiliser theorem for Lie groups;
• familiarity with some advanced topics in the theory of Lie groups such as their representation theory, symmetric spaces and holonomy.

Transferable skills

Students who have successfully taken the module will be equipped to use Lie groups in a variety of situations where they appear in Physics, Algebra, Analysis, Geometry, Number Theory or Topology. They will have background to carry research in the area or to undertake further, more advance studies on the subject. More generally, they will have had the opportunity to develop their analytic skills through the study of Lie theory, a unique blend of Algebra, Analysis and Geometry,