  By solving a set of recursion relations for the matrix elements of the ${\cal
U}_h(sl(2))$ 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 ${\cal U}_h(sl(2))$
algebra, we obtained ${\cal U}_h(so(4))$ algebra. The latter, on contraction,
yields ${\cal U}_h(e(3))$ algebra. A nonlinear map of ${\cal U}_h(e(3))$
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.
