Root systems of Levi type for Lie algebras of affine type

Let g = g(\(A\)) be a Kac--Moody Lie algebra with generalized Carlan matrix \(A\). Brundan, Goodwin and independently Kostant developed a theory of root system known as Levi type root system when \(A\) is a Carlan matrix so that g(\(A\)) is a finite dimensional semisimple Lie algebra. This theory replicates much of the structure of usual root systems. In this thesis we build up the theory of Lie algebras to review this. Then we go on to define Levi type roots for the case where \(A\) is of affine type. To describe Levi type root systems we show how these roots are related to the roots of centralizers of nilpotent elements in g. We also determine the normalizers of parabolic subgroups of finite and affine Weyl groups of classical types which can be viewed as the Weyl groups for so called root systems