Cancellation and skew cancellation for Poisson algebras

TL;DR

This paper investigates the Zariski cancellation problem for three-variable Poisson algebras, establishing conditions under which they are cancellative and exploring invariants related to skew cancellation.

Contribution

It proves that Poisson algebras with quadratic brackets or derived from Lie algebras are cancellative and introduces invariants to analyze skew cancellation.

Findings

Poisson algebras with quadratic brackets are cancellative

Poisson algebras derived from Lie algebras are cancellative

Poisson invariants help study skew cancellation

Abstract

We study the Zariski cancellation problem for Poisson algebras in three variables. In particular, we prove those with Poisson bracket either being quadratic or derived from a Lie algebra are cancellative. We also use various Poisson algebra invariants, including the Poisson Makar-Limanov invariant, the divisor Poisson subalgebra, and the Poisson stratiform length, to study the skew cancellation problem for Poisson algebras.