Poisson brackets on some skew PBW extensions

In [1] the author gives a description of Poisson brackets on some algebras of quantum polynomials \(\mathcal{O}_q\), which is called the general algebra of quantum polynomials. The main of this paper is to present a generalization of [1] through a description of Poisson brackets on some skew PBW extensions of a ring \(A\) by the extensions \(\mathcal{O}_{q,\delta}^{r,n}\), which are generalization of \(\mathcal{O}_q\), and show some examples of skew PBW extension where we can apply this description.