Simple n-Lie Poisson Algebras

Abstract

Let $(A,\cdot ,\ddot{I}‰)$ 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 $(A,\ddot{I}‰)$.