  We give functorial recipes to get, out of any Hopf algebra over a field, two
pairs of Hopf algebras which have some geometrical content. When the ground
field has characteristic zero, the first pair is made by a function algebra
over a connected Poisson group and a universal enveloping algebra over a Lie
bialgebra. In addition, the Poisson group as a variety is an affine space, and
the Lie bialgebra as a Lie algebra is graded. Forgetting these last details,
the second pair is of the same type.
  When the Hopf algebra H we start from is already of geometric type the result
involves Poisson duality: the first Lie bialgebra associated to H = F[G] is g^*
(with g := Lie(G)), and the first Poisson group H = U(g) is of type G^*, i.e.
it has g as cotangent Lie bialgebra.
  If the ground field has positive characteristic, then the same recipes give
similar results, but for the fact that the Poisson groups obtained have
dimension 0 and height 1, and restricted universal enveloping algebras are
obtained.
  We show how these "geometrical" Hopf algebras are linked to the original one
via 1-parameter deformations, and explain how these results follow from quantum
group theory. The cases of hyperalgebras and group algebras are examined in
some detail (thus recovering some well-known, classical construction), along
with some relevant examples.

Quantum Algebra;Rings and Algebras