# MA4E0 Lie Groups

**Lecturer:** Dmitriy Rumynin

**Term(s):** Term 2

**Status for Mathematics students:** List C

**Commitment:** 30 Lectures

**Assessment:** 100% 3 hour exam

**Formal registration prerequisites: **None

**Assumed knowledge: **

- MA260 Norms, Metrics and Topologies or MA222 Metric Spaces : topological spaces
- MA259 Multivariable Calculus : calculus of several variables including the Implicit Function and Inverse Function Theorems
- MA3H5 Manifolds : knowledge of manifolds, tangent spaces and vector fields will help, although all necessary results from Manifolds will be reviewed in this course

**Useful background:** A knowledge of calculus of several variables including the Implicit Function and Inverse Function Theorems, as well as the existence theorem for ODEs. A basic knowledge of manifolds, tangent spaces and vector fields will help. Results needed from the theory of manifolds and vector fields will be stated but not proved in the course.

- MA254 Theory of ODEs : the existence theorem for ODEs
- MA3F1 Introduction to Topology : homotopy groups will play a role in the later parts of this course

**Synergies: **Lie groups have both algebraic and geometric sides. These sides are studied deeply in the following two modules:

**Content**: 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.

In this module we shall introduce the classical examples of Lie groups and basic properties of the associated Lie algebra and exponential map.

**Books**:

The lectures will not follow any particular book and there are many in the Library to choose from. See section QA387. Some examples:

C. Chevalley, *Theory of Lie Groups, Vol I*, Princeton.

J.J. Duistermaat, J.A.C. Kölk, *Lie Groups*, Springer, 2000.

F.W. Warner, *Foundations of Differentiable Manifolds and Lie Groups*, (Graduate Texts in Mathematics), Springer, 1983.