  Nonlinear $sl(2)$ algebras subtending generalized angular momentum theories
are studied in terms of undeformed generators and bases. We construct their
unitary irreducible representations in such a general context. The linear
$sl(2)$-case as well as its $q$-deformation are easily recovered as specific
examples. Two other physically interesting applications corresponding to the
so-called Higgs and quadratic algebras are also considered. We show that these
two nonlinear algebras can be equipped with a Hopf structure.
