  We derive necessary and sufficient conditions for an ambiskew polynomial ring
to have a Hopf algebra structure of a certain type. This construction
generalizes many known Hopf algebras, for example U(sl2), U_q(sl2) and the
enveloping algebra of the 3-dimensional Heisenberg Lie algebra. In a
torsion-free case we describe the finite-dimensional simple modules, in
particular their dimensions and prove a Clebsch-Gordan decomposition theorem
for the tensor product of two simple modules. We construct a Casimir type
operator and prove that any finite-dimensional weight module is semisimple.
