  Let R be an integral domain, h non-zero in R such that R/hR is a field, and
HA the category of torsionless (or flat) Hopf algebras over R. We call any H in
HA "quantized function algebra" (=QFA), resp. "quantized (restricted) universal
enveloping algebra" (=QrUEA), at h if H/hH is the function algebra of a
connected Poisson group, resp. the (restricted, if R/hR has positive
characteristic) universal enveloping algebra of a (restricted) Lie bialgebra.
  We establish an "inner" Galois' correspondence on HA, via the definition of
two endofunctors, ()^\vee and ()', of HA such that:
  (a) the image of ()^\vee, resp. of ()', is the full subcategory of all
QrUEAs, resp. all QFAs, at h; (b) if R/hR has zero characteristic, the
restriction of ()^\vee to QFAs and of ()' to QrUEAs yield equivalences inverse
to each other; (c) if R/hR has zero characteristic, starting from a QFA over a
Poisson group, resp. from a QrUEA over a (restricted) Lie bialgebra, the
functor ()^\vee, resp. ()', gives a QrUEA, resp. a QFA, over the dual Lie
bialgebra, resp. the dual Poisson group.
  In particular, (a) yields a recipe to produce quantum groups of both types
(QFAs or QrUEAs), (b) gives a characterization of them within HA, and (c) gives
a "global" version of the "quantum duality principle" after Drinfeld. We then
apply our result to Hopf algebras defined over a field k and extended to the
polynomial ring k[h]: this yields quantum groups, hence "classical" geometrical
symmetries of Poisson type (via specialization) associated to the "generalized
symmetry" encoded by the original Hopf algebra over k. Both the main result and
the above mentioned application are illustrated via several examples of many
different kinds, which are studied in full detail.
