Simple n-Lie Poisson Algebras

TL;DR

The paper addresses a structural simplicity criterion for $n$-Lie Poisson algebras in characteristic $0$ with $1\in A$. It develops the $n$-ary Poisson framework, introducing $Z$, $A^{[i]}$, and adjoint derivations $\mathrm{ad}(a_2,\dots,a_n)$, and leverages a collection of lemmas (recasting AA19) to control ideals via nilpotent adjoint actions. The main result shows that $A^{[1]}/(A^{[1]}\cap Z)$ is simple whenever $(A,\cdot,\omega)$ is simple, extending analogous theorems for related $n$-Lie structures. The approach clarifies how the center governs the structure of derived $n$-Lie ideals and provides a tool for classifying simple generalized $n$-Lie-Poisson algebras and related linearly compact cases.

Abstract

Let $(A,\cdot,ω)$ 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 $(A,ω)$.

Theorems & Definitions (9)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8