Universal enveloping algebras of Poisson Hopf algebras

For a Poisson algebra $A$, by exploring its relation with Lie-Rinehart algebras, we prove a Poincar\'e-Birkoff-Witt theorem for its universal enveloping algebra $A^e$. Some general properties of the universal enveloping algebras of Poisson Hopf algebras are studied. Given a Poisson Hopf algebra $B$, we give the necessary and sufficient conditions for a Poisson polynomial algebra $B[x; \alpha, \delta]_p$ to be a Poisson Hopf algebra. We also prove a structure theorem for $B^e$ when $B$ is a pointed Poisson Hopf algebra. Namely, $B^e$ is isomorphic to $B#_\sigma \mathcal{H}(B)$, the crossed product of $B$ and $\mathcal{H}(B)$, where $\mathcal{H}(B)$ is the quotient Hopf algebra $B^e/B^eB^+$.