Degenerations of Sklyanin algebra and Askey-Wilson polynomials

TL;DR

This paper introduces a new trigonometric degeneration of the Sklyanin algebra, explores its polynomial representations, and connects it to Askey-Wilson polynomials through quadratic form diagonalization.

Contribution

It presents a novel degeneration of the Sklyanin algebra, studies its polynomial representations, and links it to Askey-Wilson polynomials via difference equations.

Findings

Degeneration leads to a subalgebra isomorphic to quantum sphere functions.

Functional realization yields difference equations for Askey-Wilson polynomials.

Contraction results in the standard quantum algebra U_q(sl(2)).

Abstract

A new trigonometric degeneration of the Sklyanin algebra is found and the functional realization of its representations in space of polynomials in one variable is studied. A further contraction gives the standard quantum algebra $U_{q}(sl(2))$. It is shown that the degenerate Sklyanin algebra contains a subalgebra isomorphic to algebra of functions on the quantum sphere $(SU(2)/SO(2))_{q^{1\over2}}$. The diagonalization of general quadratic form in its generators leads in the functional realization to the difference equation for Askey-Wilson polynomials.