Unitary representations of the quantum algebra suq(2) on a real two-dimensional sphere for q∈R+ or generic q∈S1

Abstract

Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra suq(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be expressed in terms of q-Vilenkin functions, and are related to little q-Jacobi functions, q-spherical functions, and q-Legendre polynomials. In their study, the values of q were implicitly restricted to q∈R+. In the present paper, we extend their work to the case of generic values of q∈S1 (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, q∈R+ and generic q∈S1, by determining some appropriate scalar products. From the latter, we deduce the orthonormality relations satisfied by the q-Vilenkin functions.