Dual Lie Bialgebra Structures of Poisson Types

TL;DR

This paper explores dual Lie bialgebra structures of polynomial algebras with Poisson brackets, revealing their equivalence and introducing five new classes of infinite-dimensional Lie algebras.

Contribution

It establishes the equivalence of maximal good subspaces induced by multiplication and Poisson brackets, and constructs new infinite-dimensional Lie algebras from these structures.

Findings

Maximal good subspaces coincide for multiplication and Poisson brackets.

Dual Lie bialgebra structures of Poisson type are characterized.

Five new classes of infinite-dimensional Lie algebras are constructed.

Abstract

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.