  The celebrated Milnor-Moore theorem and the more general
Cartier-Kostant-Milnor-Moore theorem establish close relationships of a
connected and a pointed cocommutative Hopf algebra with its Lie algebra of
primitive elements and its group of group-like elements. Crossed homomorphisms
for Lie algebras, groups and Hopf algebras have been studied extensively, first
from a cohomological perspective and then more broadly, with an important case
given by difference operators. This paper shows that the relationship among the
different algebraic structures captured in the Milnor-Moore theorem can be
strengthened to include crossed homomorphisms and differenece operators. We
give a graph characterization of Hopf algebra crossed homomorphisms which are
also compatible with the Milnor-Moore relation. We further investigate derived
actions from crossed homomorphisms on groups, Lie algebras and Hopf algebras,
and establish their relationship. Finally we obtain a
Cartier-Kostant-Milnor-Moore type structure theorem for pointed cocommutative
difference Hopf algebras. Examples and classifications of difference operators
are also provided for several Hopf algebras.
