  Let $k$ be an algebraically closed field of characteristic zero and let $H$
be a noetherian cocommutative Hopf algebra over $k$. We show that if $H$ has
polynomially bounded growth then $H$ satisfies the Dixmier-Moeglin equivalence.
That is, for every prime ideal $P$ in ${\rm Spec}(H)$ we have the equivalences
$$P~{\rm primitive}\iff P~{\rm rational}\iff P ~{\rm locally~closed~in}~{\rm
Spec}(H).$$ We observe that examples due to Lorenz show that this does not hold
without the hypothesis that $H$ have polynomially bounded growth. We
conjecture, more generally, that the Dixmier-Moeglin equivalence holds for all
finitely generated complex noetherian Hopf algebras of polynomially bounded
growth.
