On ${\mathcal{U}}_h(sl(2))$, ${\mathcal{U}}_h(e(3))$ and their Representations

Abstract

By solving a set of recursion relations for the matrix elements of the ${\mathcal{U}}_h(sl(2))$ generators, the finite dimensional highest weight representations of the algebra were obtained as factor representations. Taking a nonlinear combination of the generators of the two copies of the ${\mathcal{U}}_h(sl(2))$ algebra, we obtained ${\mathcal{U}}_h(so(4))$ algebra. The latter, on contraction, yields ${\mathcal{U}}_h(e(3))$ algebra. A nonlinear map of ${\mathcal{U}}_h(e(3))$ algebra on its classical analogue $e(3)$ was obtained. The inverse mapping was found to be singular. It signifies a physically interesting situation, where in the momentum basis, a restricted domain of the eigenvalues of the classical operators is mapped on the whole real domain of the eigenvalues of the deformed operators.

19 pages, latex-file, version to be published in Int. Jour. Mod. phys

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