Description
The aim of this project is to understand the classification of finite dimensional simple Lie algebras over the complex numbers. Simple Lie algebras are of fundamental importance in group theory, representation theory, number theory and mathematical physics, but to begin to understand them we only need a bit of (mainly linear) algebra.
The main idea in the classification is to associate a 'root system' to each simple Lie algebra; this is a finite set of vectors in a real vector space with many nice geometric properties (for example, it is invariant under certain reflections). Here are pictures of two of the simplest root systems. The one on the left is called $A_2$ and the one on the right is called $G_2$. They are each generated by two vectors $\alpha$ and $\beta$:
together with five "exceptional" types
$$E_6,E_7,E_8, F_4, G_2.$$
These correspond to the Dynkin diagrams:
Next we study the theory of Lie algebras over the complex numbers. The aim here is to understand how to construct a root system from a simple Lie algebra. We then study the main theorem, which states that the classification of irreducible root systems provides a classification of the simple Lie algebras.
Finally, there are several possible further directions to investigate individually: Finite simple groups, linear algebraic groups, Witten zeta functions, etc.
Prerequisites
Algebra IIResources
- R. Carter, G. Segal, I. Macdonald, Lectures on Lie
Groups and Lie Algebras (Carter's chapter is a
very readable introduction and gives a good overview but
without proofs)
- R. Carter, Lie Algebras of Finite and Affine Type
- K. Erdmann and M. Wildon, Introduction to Lie Algebras