Simple n-Lie Poisson Algebras

TL;DR

This paper investigates the structure of simple n-Lie Poisson algebras, proving that a certain quotient of the derived ideal is simple, which advances understanding of their algebraic properties.

Contribution

It establishes the simplicity of the quotient of the derived n-Lie ideal over the intersection with the center in simple n-Lie Poisson algebras.

Findings

The quotient $A^{[1]}/(A^{[1]}\cap Z)$ is simple.

Provides structural insights into simple n-Lie Poisson algebras.

Extends the theory of n-Lie algebra simplicity.

Abstract

Let $(A,\cdot,\omega)$ be a simple $n$-Lie Poisson algebra over a field of zero characteristic, $ 1 \in A.$ Then we prove that the $n$-Lie algebra $A^{[1]}/(A^{[1]}\cap Z)$ is simple, where $A^{[1]}$ denotes the derived $n$-Lie ideal and $Z$ is the center of $n$-Lie algebra $(A,\omega)$.