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Degenerations of Sklyanin algebra and Askey-Wilson polynomials

A. S. Gorsky·A. V. Zabrodin
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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.

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A. S. Gorsky, A. V. Zabrodin. 1993-03-04. Degenerations of Sklyanin algebra and Askey-Wilson polynomials. https://doi.org/10.1088/0305-4470%2F26%2F15%2F004

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