MA4E0 Lie Groups
Lecturer: Weiyi Zhang
Term(s): Term 2
Status for Mathematics students: List C
Commitment: 30 Lectures
Assessment: 100% 3 hour exam
Formal registration prerequisites: None
Assumed knowledge:
- 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.