Dual Lie bialgebra structures of Poisson types

Abstract

Let $\mathcal{A} = \mathbb{F}[x,y]$ be the polynomial algebra on two variables x, y over an algebraically closed field $\mathbb{F}$ of characteristic zero. Under the Poisson bracket, $\mathcal{A}$ is equipped with a natural Lie algebra structure. It is proven that the maximal good subspace of $\mathcal{A}*$ induced from the multiplication of the associative commutative algebra $\mathcal{A}$ coincides with the maximal good subspace of $\mathcal{A}*$ induced from the Poisson bracket of the Poisson Lie algebra $\mathcal{A}$ . 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.