  The Hopf dual $H^\circ$ of any Poisson Hopf algebra $H$ is proved to be a
co-Poisson Hopf algebra provided $H$ is noetherian. Without noetherian
assumption, it is not true in general. There is no nontrivial Poisson Hopf
structure on the universal enveloping algebra of a non-abelian Lie algebra. The
Poisson Hopf structures on $A=k[x_1, x_2, \cdots, x_d]$, viewed as the
universal enveloping algebra of a finite-dimensional abelian Lie algebra, are
exactly linear Poisson structures on $A$. The co-Poisson structures on
polynomial Hopf algebra $A$ are characterized. Some correspondences between
co-Poisson and Poisson structures are also established.
