Poisson brackets on some skew PBW extensions

TL;DR

This paper generalizes the description of Poisson brackets from quantum polynomial algebras to skew PBW extensions, broadening the understanding of algebraic structures with Poisson geometry.

Contribution

It extends previous work by providing a framework for Poisson brackets on skew PBW extensions, including new examples and applications.

Findings

Poisson brackets described on new classes of skew PBW extensions

Generalization of quantum polynomial algebra results

Examples illustrating the application of the theory

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.