Universal coacting Poisson Hopf algebras

TL;DR

This paper constructs a universal coacting Poisson Hopf algebra for Poisson algebras, extending Manin's concept to the Poisson setting, and establishes its universal properties and structures.

Contribution

It introduces the universal coacting Poisson Hopf algebra for Poisson algebras, providing a new framework and universal properties in the Poisson algebra context.

Findings

Constructed a Poisson algebra al B(P, U) with universal property.

Showed al B(P, U) admits a Poisson bialgebra structure.

Defined the universal coacting Poisson Hopf algebra al H(P) as an initial object.

Abstract

We introduce the analogue of Manin's universal coacting (bialgebra) Hopf algebra for Poisson algebras. First, for two given Poisson algebras $P$ and $U$, where $U$ is finite dimensional, we construct a Poisson algebra $\mathcal{B}(P,\, U)$ together with a Poisson algebra homomorphism $\psi_{\mathcal{B}(P,\,U)} \colon P \to U \otimes \mathcal{B}(P,\, U)$ satisfying a suitable universal property. $\mathcal{B}(P,\, U)$ is shown to admit a Poisson bialgebra structure for any pair of Poisson algebra homomorphisms subject to certain compatibility conditions. If $P=U$ is a finite dimensional Poisson algebra then $\mathcal{B}(P) = \mathcal{B}(P,\, P)$ admits a unique Poisson bialgebra structure such that $\psi_{\mathcal{B}(P)}$ becomes a Poisson comodule algebra and, moreover, the pair $\bigl(\mathcal{B}(P),\, \psi_{\mathcal{B}(P)}\bigl)$ is the universal coacting bialgebra of $P$. The universal coacting Poisson Hopf algebra $\mathcal{H}(P)$ on $P$ is constructed as the initial object in the category of Poisson comodule algebra structures on $P$ by using the free Poisson Hopf algebra on a Poisson bialgebra (\cite{A1}).