Log-ozone groups and centers of polynomial Poisson algebras

TL;DR

This paper introduces the log-ozone group of Poisson algebras in positive characteristic and uses it to characterize polynomial Poisson algebras with skew symmetric structures, revealing properties of their centers.

Contribution

It extends the concept of ozone groups to log-ozone groups in Poisson algebras and characterizes polynomial Poisson algebras with skew symmetric structures.

Findings

Unimodular Poisson algebras with skew symmetric structure have Gorenstein centers.

Log-ozone groups help classify polynomial Poisson algebras in positive characteristic.

Results include properties of graded polynomial Poisson algebras of dimension three.

Abstract

In previous work, the authors introduced the ozone group of an associative algebra as the subgroup of automorphisms which fix the center pointwise. The authors studied PI skew polynomial algebras, using the ozone group to understand their centers and to characterize them among graded algebras. In this work, we introduce and study the log-ozone group of a Poisson algebra over a field of positive characteristic. The log-ozone group is then used to characterize polynomial Poisson algebras with skew symmetric structure. We prove that unimodular Poisson algebras with skew symmetric structure have Gorenstein centers. A related result is proved for graded polynomial Poisson algebras of dimension three.