  We prove an algebraic ``no-go theorem'' to the effect that a nontrivial
Poisson algebra cannot be realized as an associative algebra with the
commutator bracket. Using this, we show that there is an obstruction to
quantizing the Poisson algebra of polynomials generated by a nilpotent basic
algebra on a symplectic manifold. Finally, we explicitly construct a polynomial
quantization of a symplectic manifold with a solvable basic algebra, thereby
showing that the obstruction in the nilpotent case does not extend to the
solvable case.
