Derivations of the Lie algebra of polynomials under Poisson bracket.

We exhibit a class of outer derivations of the Lie algebra $P$ of complex polynomials under Poisson bracket, and prove that every derivation of $P$ is a linear combination of one of these and an inner derivation, although this decomposition may not be unique. In particular, we show that any derivation of $P$ which maps constants to zero must be inner. We use these results to characterise certain solutions of the Dirac problem.