Unitary representations of the quantum algebra su_q(2) on a real two-dimensional sphere for $q\in R^{+}$ or generic $q\in S^1$

Abstract

Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra su_q(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 $q\in R^{+}$. In the present paper, we extend their work to the case of generic values of $q\in S^1$ (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, $q\in R^{+}$ and generic $q\in S^1$, by determining some appropriate scalar products. From the latter, we deduce the orthonormality relations satisfied by the q-Vilenkin functions.

24 pages, LaTeX, uses amssym, 1 PostScript figure

Cited by in corpus (1)

• Unitary Representations of Su_q(2) on the Plane for q in R+ or Generic q in S1