DEGENERATIONS OF SKLYANIN ALGEBRA AND ASKEY-WILSON POLYNOMIALS

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))q1/2. The diagonalization of general quadratic form in its generators leads in the functional realization to the difference equation for Askey-Wilson polynomials.