Deformation quantization of polynomial Poisson algebras
An explicit third order deformation quantization of an arbitrary polynomial Poisson algebra is provided. The second and third order terms are obtained formally in the same way as analogous terms of the star-product induced by the universal enveloping algebra in the linear case. For instance, the second term for the Poisson bracket \(\pi_1=\frac{1}{2}X_{ij} \partial^i\wedge\partial^j\) reads \[ \pi_2=\frac{1}{6}X^l_{ij}X_{lk}\partial^i\cdot\partial^{jk}+ \frac{1}{8}X_{ij}X_{kl}\partial^{ik}\cdot\partial^{jl}, \] where \(\cdot\) is the tensor symmetrization. This deformation quantization differs from the Kontsevich one and not always extends to the order four. A correction term is computed which cancels this obstruction.
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