Simple Poisson modules over a Poisson algebra $S_q$

Nagornchat Chansuriya, Nongkhran Sasom

Abstract

Let $S$ be a $\mathbb{C}$-algebra generated by $x,\, y,\, z,\,q$ and$q^{-1}$ satisfies the relations\begin{eqnarray*} \label{xy1}xy - qyx &=&(q-1)(x+y+z),\\ \label{yz1}yz-qzy &=& (q-1)(x+y+z),\\ \label{zx1}zx-qxz &=& (q-1)(x+y+z) \quad \mbox{  and }\end{eqnarray*}\[\label{centrt}xq=qx,\quad yq=qy,\quad zq =qz,\quadqq^{-1}=1=q^{-1}q.\] 
We focus on a Poisson algebra $S_q$, constructed from $S$, with Poisson brackets $\{x,y\} = yx+x+y+z,$ $\{y,z\} = zy +x+y+z,$ and $\{z,x\}= xz + x+y+z.$   There are only two Poisson maximal ideals of $S_q$  . In this study, we characterize the simple Poisson modules which annihilated by each of the Poisson maximal ideals of $S_q$ .