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.

Qmechanic · Answer

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
$$begin{align}fstar g~:=~& f expleft(stackrel{leftarrow}{partial_I}frac{ihbar}{2} omega^{IJ} stackrel{rightarrow}{partial_J}right) gcr 
~=~&fg+frac{ihbar}{2}{f,g}_{PB} +{cal O}(hbar^2). end{align}tag{2} $$
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.

Quantization of an $mathcal{c}$-algebra

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