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.

The Racah polynomial Rn(λ(x)) is a polynomial of degree n and is variable in λ(x). In this thesis two properties of this polynomial will be studied. One is the orthogonal property of the Racah polynomial. And the other is that the Racah polynomial can also be described as a polynomial of degree x and variable over λ(n). The Racah polynomials will be studied with the use of a representation of the Lie algebra of sl(2;C) and hypergeometric series. To do this, this Lie algebra will first be defined and then we will work towards defining the tensor product of three representations of the Lie algebra sl(2;C). From the tensor product, a series representation for the Racah polynomials will be found, which can be rewritten to a hypergeometric series. Then, the orthogonal property of sl(2;C) will be used to study the orthogonal property of the Racah polynomials. And the polynomial will be rewritten as a polynomial of degree x with the use of some identities of the hypergeometric series.