Quantization of an $\mathcal{c}$-algebra

Question

I don't know if this is a valid question to ask, but I am wondering about the following: We are given a set of $\mathcal{c}$-number Lie-Brackets

$$ [q_i,q_j] = 0= [p_i,p_j] \\ [q_i,p_j] = c_{ij}, $$

with $c_{ij} \in \mathbb{C}$. Is it possible to obtain a quantum theory from this algebra? If yes, is there some kind of recipe how this is done? If no, why not, and what's the best way to approach such a question?

This post is related to Quantization of $c$-number Dirac-Bracket, but kept more general.

Answer 1

For what it's worth, given a $z$-independent Poisson structure $$\begin{align}\{z^I,z^J\}_{PB}~=~&\omega^{IJ}~\in~\mathbb{R}, \cr I,J~\in~&\{1,\ldots,2n\}, \end{align}\tag{1} $$ where $\omega^{IJ}$ is an arbitrary (not necessarily invertible) real skewsymmetric matrix, one may define an associative non-commutative Groenewold-Moyal star product $$\begin{align}f\star g~:=~& f \exp\left(\stackrel{\leftarrow}{\partial_I}\frac{i\hbar}{2} \omega^{IJ} \stackrel{\rightarrow}{\partial_J}\right) g\cr ~=~&fg+\frac{i\hbar}{2}\{f,g\}_{PB} +{\cal O}(\hbar^2). \end{align}\tag{2} $$ The functions/symbols $f,g$ with the star product $\star$ is a representation of corresponding operators $\hat{f},\hat{g}$ with the composition $\circ$, cf. e.g. my Phys.SE answer here. It's a quantization in that sense.