Word problem for finitely presented metabelian Poisson algebras

TL;DR

This paper develops a basis construction and Gröbner--Shirshov methods for metabelian Poisson algebras, proving the word problem for finitely presented cases is solvable, with results depending on the field's characteristic.

Contribution

It introduces a basis for free metabelian Poisson algebras, adapts Gröbner--Shirshov techniques, and solves the word problem for finitely presented algebras.

Findings

Constructed a linear basis depending on field characteristic

Developed Gröbner--Shirshov basis method for these algebras

Proved the word problem is solvable for finitely presented cases

Abstract

We first construct a linear basis for a free metabelian Poisson algebra generated by an arbitrary well-ordered set. It turns out that such a linear basis depends on the characteristic of the underlying field. Then we elaborate the method of Gr\"{o}bner--Shirshov bases for metabelian Poisson algebras. Finally, we show that the word problem for finitely presented metabelian Poisson algebras are solvable.