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29 March 2024
 
  » arxiv » math.QA/9808091

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Unitary representations of the quantum algebra su_q(2) on a real two-dimensional sphere for $q in R^+$ or generic $q in S^1$
M. Irac-Astaud ; C. Quesne ;
Date 20 Aug 1998
Journal J. Math. Phys. 40 (1999) 3146-3161
Subject Quantum Algebra | math.QA
AbstractSome 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.
Source arXiv, math.QA/9808091
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