Derivations of the Lie algebra of polynomials under Poisson bracket.

Abstract

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 con- stants to zero must be inner. We use these results to characterise certain solutions of the Dirac problem. 1. Introduction. Let E denote the collection of all C* complex- valued functions of 2n real variables x = (ql, * , qn Pig, , Pn)i and define the Poisson bracket of two elements of E to be {j,g}~~9cf Og Of 49g)