The Clebsch-Gordan coefficients for a family of natural modules of the Modular Double of the quantum group Uq(sl(2,R))

Abstract

We will study the Clebsch-Gordan coefficients of the modular double of the quantum group Uq(sl(2, R)). This will be done by studying and taking a good look at how B. Ponsot and J. Teschner showed how to compute the Clebsch-Gordan coefficients [1]. Moreover, we will also take an introductory look at the concept of quantum groups by looking at some general theory on Hopf ∗-algebras and their representations. The Clebsch-Gordan coefficients can roughly be described as a relation between a basis of a tensor product U ⊗V of two simple Uq(sl(2, R))-modules and a basis of the decomposition of U ⊗V into simple modules. We will show that this relation can be explicitly described by an integral transformation. Since this describes a relation between modules of a quantum group, the first part of this thesis will give the necessary information to introduce the reader to the concept of quantum groups and their modules. This will be done by introducing Hopf algebras and their modules and then look at their quantum deformations. This first part will also introduce several examples of algebras, Hopf algebras and quantum groups to make the reader get used to the concept of Hopf algebras and quantum groups.