  In this article we propose a new and so-called holomorphic deformation scheme
for locally convex algebras and Hopf algebras. Essentially we regard converging
power series expansion of a deformed product on a locally convex algebra, thus
giving the means to actually insert complex values for the deformation
parameter. Moreover we establish a topological duality theory for locally
convex Hopf algebras. Examples coming from the theory of quantum groups are
reconsidered within our holomorphic deformation scheme and topological duality
theory. It is shown that all the standard quantum groups comprise holomorphic
deformations. Furthermore we show that quantizing the function algebra of a
(Poisson) Lie group and quantizing its universal enveloping algebra are
topologically dual procedures indeed. Thus holomorphic deformation theory seems
to be the appropriate language in which to describe quantum groups as deformed
Lie groups or Lie algebras.
