  Let $A=F[x,y]$ be the polynomial algebra on two variables $x,y$ over an
algebraically closed field $F$ of characteristic zero. Under the Poisson
bracket, $A$ is equipped with a natural Lie algebra structure. It is proven
that the maximal good subspace of $A^*$ induced from the multiplication of the
associative commutative algebra $A$ coincides with the maximal good subspace of
$A^*$ induced from the Poisson bracket of the Poisson Lie algebra $A$. Based on
this, structures of dual Lie bialgebras of the Poisson type are investigated.
As by-products, five classes of new infinite dimensional Lie algebras are
obtained.
