Poisson-Hopf limit of quantum algebras

The Poisson-Hopf analogue of an arbitrary quantum algebra U-z(g) is constructed by introducing a one-parameter family of quantizations U-z,U-h(g) depending explicitly on (h) over bar h and by taking the appropriate (h) over bar -> 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.