# Simplicity of Lie algebras of Poisson brackets

## TL;DR

This paper proves that for a simple Poisson algebra over a field of zero characteristic, the associated Lie algebra formed by Poisson brackets modulo the center is also simple, highlighting a fundamental structural property.

## Contribution

It establishes the simplicity of the Lie algebra derived from the Poisson brackets in simple Poisson algebras, extending understanding of their algebraic structure.

## Key findings

- The Lie algebra of Poisson brackets modulo the center is simple.
- Simplicity of the Poisson algebra implies simplicity of the associated Lie algebra.
- Provides a structural link between Poisson and Lie algebra properties.

## Abstract

Let $A$ be an associative commutative algebra with $1$ over a field of zero characteristic, $\{,\} : A \times A \to A$ is a Poisson bracket, $Z = \{ a \in A \mid \{a, A\} = (0) \}.$ We prove that if $A$ is simple as a Poisson algebra then the Lie algebra $\frac{\{A,A\}}{\{A,A\}\cap Z}$ is simple.