Co-Poisson structures on polynomial Hopf algebras

TL;DR

This paper investigates co-Poisson structures on polynomial Hopf algebras, establishing duality results, characterizing structures on polynomial algebras, and exploring the correspondence between Poisson and co-Poisson structures.

Contribution

It proves the duality of co-Poisson structures on noetherian Poisson Hopf algebras and characterizes co-Poisson structures on polynomial Hopf algebras.

Findings

Duality of co-Poisson structures on noetherian Poisson Hopf algebras

Characterization of co-Poisson structures on polynomial Hopf algebras

Establishment of correspondences between Poisson and co-Poisson structures

Abstract

The Hopf dual $H^\circ$ of any Poisson Hopf algebra $H$ is proved to be a co-Poisson Hopf algebra provided $H$ is noetherian. Without noetherian assumption, it is not true in general. There is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. The Poisson Hopf structures on $A=k[x_1, x_2, \cdots, x_d]$, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, are exactly linear Poisson structures on $A$. The co-Poisson structures on polynomial Hopf algebra $A$ are characterized. Some correspondences between co-Poisson and Poisson structures are also established.