Deformation quantization of polynomial Poisson algebras

This paper discusses the notion of a deformation quantization for an arbitrary polynomial Poisson algebra A. We compute an explicit third order deformation quantization of A and show that it comes from a quantized enveloping algebra. We show that this deformation extends to a fourth order deformation if and only if the quantized enveloping algebra gives a fourth order deformation; moreover we give an example where the deformation does not extend. A correction term to the third order quantization given by the enveloping algebra is computed, which precisely cancels the obstruction, so that the modified third order deformation extends to a fourth order one. The solution is generically unique, up to equivalence. (C) 2000 Academic Press.