Quantum groups and Askey-Wilson polynomials
C.C.M.L. Wagenaar (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Wolter Groenevelt – Mentor (TU Delft - Analysis)
JMAM Neerven – Graduation committee member (TU Delft - Analysis)
DC Gijswijt – Graduation committee member (TU Delft - Discrete Mathematics and Optimization)
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Abstract
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 expression to define generalised matrix elements and spherical functions in Aq(SL(2,C)). Then we use the Haar functional to show that these generalised matrix elements are Askey-Wilson polynomials with two continuous and two discrete parameters. Next, we show a new result. Namely, two twisted primitive elements of Uq(SL(2,C)) generate Zhedanov's Askey-Wilson algebra AW(3). Consequently, AW(3) is embedded as a subalgebra into Uq(SL(2,C)). We use this to show that overlap functions of twisted primitive elements in Uq(SU(2)) are q-Racah polynomials. With that, we derive a summation formula connecting q-Racah and dual q-Krawtchouk polynomials.