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.