Poisson brackets on some skew PBW extensions

Abstract

In [1] the author gives a description of Poisson brackets on some algebras of quantum polynomials $\mathcal{O}_q$, which is called\textit{ 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 $\mathcal{O}^{r,n}_{q,\delta}$, which are a generalization of $\mathcal{O}_q$ and show some examples of skew PBW extension where we can apply this description.