the Degree of Doctor of Philosophy of the University of London Representations at a Root of Unity of q - Oscillators and Quantum Kac - Moody Algebras

The subject of this thesis is quantum groups and quantum algebras at a root of unity. After an introductory chapter, I set up my notation in chapter 2. The rest of the thesis is presented in three parts. In part I, quantum matrix groups and quantum enveloping algebras are discussed. In chapter 3, I present two well-known 2 × 2 matrix quantum groups, including their coaction on the quantum plane and specialisations at a root of unity. Chapter 4 develops a quite detailed description of quantum enveloping algebras and their specialisation at an odd root of unity. The results from this chapter are required in part III. Part II is devoted to certain deformations of the quantum mechanical oscillator algebra: so called q-oscillators. In chapter 5, a standard q-oscillator and its Fock module is described, including its specialisation at a root of unity. In chapter 6, original work [Pet93] on a new 2-parameter deformation of the oscillator algebra is presented and its representations at a root of unity are described. Part III is concerned with infinite dimensional quantum groups. In chapter 7, the structure of an (untwisted) quantum affine Kac-Moody algebra is discussed. As in the classical case, it has both a Chevalley and a loop algebra presentation, which can be shown to be isomorphic using braid group and translation automorphisms. A quantum affine algebra has also a Heisenberg subalgebra: I describe its Fock modules and their unitarisability. Finally in chapter 8, I present original results [Pet94] on the specialisation of a quantum affine algebra at an odd root of unity. I prove that a quantum affine algebra at a root of unity has an infinite dimensional centre and construct the central elements corresponding to the real and imaginary roots. At the odd root of unity, some new infinite dimensional representations of the algebra are shown to exist.