We prove that there are no nontrivial finite-dimensional Lie representations of certain Poisson algebras of polynomials on a compact symplectic manifold. This result is used to establish the existence of a universal obstruction to quantizing a compact symplectic manifold, regardless of the dimensionality of the representation.
The Addendum contains a simplified proof of Theorem 4.
7 pps. 5/30/97 (Revised 10/16/97). Published in: Proc. Amer. Math. Soc. 128, 237--243 (2000). Addendum, 2 pps., 10/14/99.