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

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...

A multivariate analogue of Ruijsenaars's generalised hypergeometric function: A quantum algebra approach

In this thesis, we derive a multivariate analogue of Ruijsenaars’s 2F1-generalisation R. We use Hopf algebra representation theory of the modular double of sl.2/, a Hopf algebra structure strongly related to quantum groups, to relate the function R to overlap coefficients of eigenfunctions. Using properties of the algebra and the representation,...

Information-theoretic quantities of quantum channels with partition quantum group symmetries

In the field of quantum information theory, it is well-known that the purely quantum phenomenon called quantum entanglement can boost the capacity of a quantum channel, which is called the superadditivity of the capacity. Shor showed in his breakthrough paper on the equivalence of additivity conjectures that this superadditivity is equivalent to...

Quantum groups and Askey-Wilson polynomials

In this thesis, we introduce the quantum groups Uq(SL(2,C)) and Aq(SL(2,C)) as Hopf algebras. We study their representations, including their similarities and differences with the classical theory. We show that the eigenvectors of Koorwinder's twisted primitive elements of Uq(SU(2)) are dual q-Krawtchouk polynomials. We use this explicit...

Riesz transforms on compact quantum groups and strong solidity

One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call 'approximate linearity with almost...