  The Poisson-Hopf analogue of an arbitrary quantum algebra U_z(g) is
constructed by introducing a one-parameter family of quantizations U_{z,h}(g)
depending explicitly on h and by taking the appropriate h -> 0 limit. The
q-Poisson analogues of the su(2) algebra are discussed and the novel su_q^P (3)
case is introduced. The q-Serre relations are also extended to the Poisson
limit. This approach opens the perspective for possible applications of higher
rank q-deformed Hopf algebras in semiclassical contexts.
