Poisson brackets of the generalized functions

Question

Let $M$ be a manifold. I define $A(I)$  the commutativ algebra of generalized functions:

$$A(I)={\cal C}^{\infty}(M)[X_1,X_2,\ldots,X_k]/I$$

where $I$ is an ideal of ${\cal C}^{\infty}(M)[X_1,X_2,\ldots,X_k]$, the polynomials over the smooth functions, such that $A(I)$ is of finite type.

Then I define  Poisson brackets:

$$\{ a,a'\}=\omega (da,da')$$

where $a,a' \in A(I)$ and $\omega \in \Lambda^2(TA(I))$ is a symplectic form of $TA(I)=Der(A(I))$ the derivations of $A(I)$.

Can we quantize the structure?