Course description:

Lie groups appear in mathematics and physics as the symmetry groups of various objects, spaces, or systems. A Lie group is a differentiable manifold with a group structure, the group operations of which are smooth maps. Thus, the general theory of Lie groups relies heavily on the theory of differentiable manifolds. On he other hand, almost all Lie groups can be represented as matrix groups, the study of which requires only some familiarity with elementary linear algebra. The goal of the course is to give the flavour of general Lie theory by presenting some fundamental theorems and constructions on the special case of matrix Lie groups, keeping prerequisites to a minimum. Understanding the classical examples serves as a motivation for the general theory and makes the abstract theory more accessible on an advanced level.

Topics covered:

• Vector fields and one-parameter groups of matrices
• The properties of the exponential map
• The Campbell—Baker—Hausdorff formula
• Linear groups
• The Lie-algebra of a linear group
• Coordinates on a linear group
• Connected groups
• The Lie correspondence
• Homomorphisms of linear groups
• Closed subgroups of linear groups
• The classical matrix groups
• Cartan subgroups, the Weyl group
• Roots, weights and reflections
• Lie groups and Lie algebras