Michel Van den Bergh introduced the notion of a double Poisson algebra. The definition is cooked up such that the representation varieties of such an algebra are Poisson varieties.
Is there a notion of "quantization", or "double star-product" for double Poisson algebras, so that it would induce genuine star-products quantizing the above mentioned Poisson varieties ?
EDIT: being more precise, if an associative algebra $A$ is equipped with a double Poisson structure, then for any $n\geq0$ the affine variety $Rep_n(A)$ of $n$-dimensional representations of $A$ is naturally equipped with a (algebraic) Poisson structure. Remark: the symplectic version of this story goes back to Crawley-Boevey, Etingof and Ginzburg, who studied representation of quivers using non-commutative symplectic geometry). Now my question is following:
What kind of algebraic structure on an algebra $A$ ensures that one will get star-products on $Rep_n(A)$. Here by a star-product on an affine variety $X$ I improperly mean a formal one-parameter deformation of its function ring $\mathcal O_X$.
