Finite-dimensional simple Poisson modules

TL;DR

This paper develops a method to classify finite-dimensional simple Poisson modules over Poisson algebras, illustrating the approach with various examples and exploring their relation to deformations and invariants under group actions.

Contribution

It introduces a general result for identifying finite-dimensional simple Poisson modules and applies it to multiple examples, highlighting connections with module deformations and invariants.

Findings

Classification method for finite-dimensional simple Poisson modules

Examples demonstrating the correspondence with deformed modules

Analysis of modules under invariants of finite group actions

Abstract

We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the finite-dimensional simple modules over deformations and on the behaviour of finite-dimensional simple Poisson modules on the passage from a Poisson algebra to the Poisson subalgebra of invariants for the action of a finite group of Poisson automorphisms.