# Invariants of Unimodular Quadratic Polynomial Poisson Algebras of Dimension 3

## TL;DR

This paper extends classical theorems like Shephard-Todd-Chevalley and Watanabe to the setting of unimodular quadratic Poisson algebras of dimension 3 and their Poisson enveloping algebras, analyzing invariants under automorphisms.

## Contribution

It proves a variant of the Shephard-Todd-Chevalley theorem for 3-dimensional unimodular quadratic Poisson algebras and extends related results to their Poisson enveloping algebras.

## Key findings

- Established a Shephard-Todd-Chevalley type theorem for the Poisson algebra.
- Extended the theorem to the Poisson enveloping algebra under induced group actions.
- Provided structural insights into invariants of unimodular quadratic Poisson algebras.

## Abstract

Let $P = \Bbbk[x1,x2,x3]$ be a unimodular quadratic Poisson algebra and let $G$ be a finite subgroup of the graded Poisson automorphism group of $P$. In this paper, we prove a variant of the Shephard-Todd-Chevalley theorem for $P$ and variants the Shephard-Todd-Chevalley theorem and the Watanabe theorem for its Poisson enveloping algebra $U(P)$ under the induced group $\widetilde{G}$.