A Groenewold-VanHove theorem for S-2

We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold S-2 which is irreducible on the su(2) subalgebra generated by the components {S-1, S-2, S-3} of the spin vector. In fact there does not exist such a quantization of the Poisson subalgebra P consisting of polynomials in {S-1, S-2, S-3}. Furthermore, we show that the maximal Poisson subalgebra of P containing {1, S-1, S-2, S-3} that can be so quantized is just that generated by {1, S-1, S-2, S-3}.