  We discuss the Groenewold-Van Hove problem for R^{2n}, and completely solve
it when n = 1. We rigorously show that there exists an obstruction to
quantizing the Poisson algebra of polynomials on R^{2n}, thereby filling a gap
in Groenewold's original proof without introducing extra hypotheses. Moreover,
when n = 1 we determine the largest Lie subalgebras of polynomials which can be
unambiguously quantized, and explicitly construct all their possible
quantizations.
