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

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

1998-08-20

arxiv.org/abs/math/9808091v1

J. Math. Phys. 40 (1999) 3146-3161

Mathematics

Quantum Algebra

24 pages, LaTeX, uses amssym, 1 PostScript figure

Scientific paper

10.1063/1.532751

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.

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