  The "quantum duality principle" states that a quantisation of a Lie bialgebra
provides also a quantisation of the dual formal Poisson group and, conversely,
a quantisation of a formal Poisson group yields a quantisation of the dual Lie
bialgebra as well. We extend this to a much more general result: namely, given
any principal ideal domain R, for each prime h in R we establish sort of an
"inner" Galois' correspondence on the category HA of torsionless Hopf algebras
over R, via the definition of two functors (from HA to itself) such that the
image of the first, resp. of the second, is the full subcategory of those Hopf
algebras which are commutative, resp. cocommutative, modulo h (i.e. they are
"quantum function algebras" (=QFA), resp. "quantum universal enveloping
algebras" (=QUEA), at h). In particular we provide a machine to get two quantum
groups - a QFA and a QUEA - out of any Hopf algebra H over a field k: just plug
in a parameter x and apply the functors to H[x] for h = x. Several relevant
examples are studied in full detail: the trivial quantisations, the semisimple
groups, the Euclidean group, the Heisenberg group, and the Kostant-Kirillov
structure on any Lie algebra; furthermore, an interesting application to
renormalisation theory in quantum electro-dynamics is studied, as a sample of
application of the principle to a quite large class of problems.
  This work is a far-reaching "evolution" of the same author's preprint
math.QA/9912186: the present paper is entirely self-contained, is more general
from the mathematical point of view, and contains additional examples.
  WARNING: This preprint has been overtaken by a new, deeply enhanced and
improved version, available as {\tt math.QA/0303019}; the interested reader is
kindly asked to refer to that new preprint.
