Bulletin of the
Korean Mathematical Society BKMS

Let $A$ be a co-Poisson Hopf algebra with Poisson co-bracket $\delta$. Here it is shown that the Hopf dual $A^\circ$ is a Poisson Hopf algebra with Poisson bracket $\{f,g\}(x)=\langle\delta(x), f\otimes g\rangle$ for any $f,g\in A^\circ$ and $x\in A$ if $A$ is an almost normalizing extension over the ground field. Moreover we get, as a corollary, the fact that the Hopf dual of the universal enveloping algebra $U(\frak g)$ for a finite dimensional Lie bialgebra $\frak g$ is a Poisson Hopf algebra.

Keywords: co-Poisson Hopf algebra, Poisson Hopf algebra

MSC numbers: 17B62, 17B63, 16W30