Simplicity of Lie algebras of Poisson brackets

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.

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