Star-product of a quadratic Poisson structure (Q675777)

Using the results of \textit{J. Grabowski} [J. Geom. Phys. 9, No. 1, 45--73 (1992; Zbl 0761.16012)] and \textit{H. Omori}, \textit{Y. Maeda} and \textit{A. Yoshioka} [Contemp. Math. 179, 213--240 (1994; Zbl 0820.58027)] on the existence of a star-product of a Poisson structure, we show that ``generic'' quadratic Poisson structures in dimension \(n\) [\textit{J.-P. Dufour} and \textit{A. Haraki}, C. R. Acad. Sci., Paris, Sér. I 312, No. 1, 137--140 (1991; Zbl 0719.58001) and \textit{Z.-J. Liu} and \textit{P. Xu}, Lett. Math. Phys. 26, No. 1, 33--42 (1992; Zbl 0773.58007)] do admit star-products and, in dimension 3, every quadratic Poisson structure does admit a star-product. Examples of star-products are given for the 13 first classes of the classification of Dufour-Haraki [loc. cit.]. For the model \#14 (the last one of the previous classification), we give an example of star-product when the polynomials \(Q\) (that generate this case) are the product of two non-constant polynomials.