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

Abstract

By solving a set of recursion relations for the matrix elements of the [Formula: see text] 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 [Formula: see text] algebra, we obtained [Formula: see text] algebra. The latter, on contraction, yields [Formula: see text] algebra. A nonlinear map of [Formula: see text] algebra on its classical analog 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.