Universal enveloping algebras of Poisson Hopf algebras

For a Poisson algebra A  , by exploring its relation with Lie–Rinehart algebras, we prove a Poincaré–Birkhoff–Witt theorem for its universal enveloping algebra AeAe. 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;α,δ]pB[x;α,δ]p to be a Poisson Hopf algebra. We also prove a structure theorem for BeBe when B   is a pointed Poisson Hopf algebra. Namely, BeBe is isomorphic to B#σH(B)B#σH(B), the crossed product of B   and H(B)H(B), where H(B)H(B) is the quotient Hopf algebra Be/BeB+Be/BeB+.