Defining relations and Gr\"obner--Shirshov bases of Poisson algebras as of conformal modules

TL;DR

This paper explores the connection between Poisson algebras and Lie conformal algebra representations, developing a method to compute Gr"obner--Shirshov bases in Poisson algebras viewed as conformal modules.

Contribution

It introduces a new framework for calculating Gr"obner--Shirshov bases in Poisson algebras using conformal module techniques over associative envelopes.

Findings

Established a setting for Gr"obner--Shirshov basis calculation in Poisson algebras.

Linked Poisson algebra structures to conformal modules over associative algebras.

Provided a computational approach for Poisson algebra analysis.

Abstract

We study the relation between Poisson algebras and representations of Lie conformal algebras. We establish a setting for the calculation of a Gr\"obner--Shirshov basis in a module over an associative conformal algebra and apply this technique to Poisson algebras considered as conformal modules over appropriate associative envelopes of current Lie conformal algebras. As a result, we obtain a setting for the calculation of a Gr\"obner--Shirshov basis in a Poisson algebra.