Jordanian presentation of the Weyl algebra \(A_2\) (Q2724114)

Let \(k\) be a field and \(a,b\in k\). Suppose that \(J_{a,b}\) is the Weyl algebra on a Jordan plane, that is an associative algebra with generators \(x_1,x_2,\partial_1,\partial_2\) subject to defining relations NEWLINE\[NEWLINE\begin{alignedat}{2} 3[x_1,x_2]&=ax_1^2,&\qquad[\partial_1,x_1]&=1+ax_1\partial_2,&\qquad[\partial_1,\partial_2]&=-b\partial_2^2,\\ [\partial_2,x_2]&=1-bx_1\partial_2,&\qquad[\partial_2,x_1]&=0,&\qquad[\partial_1,x_2]&=-ax_1\partial_1-abx_1\partial_2+bx_2\partial_2.\end{alignedat}NEWLINE\]NEWLINE It is observed that the element \(z=1+(a-b)x_1\partial_2\) is normal. Denote by \(H_{a,b}\) the localization of \(J_{a,b}\) by the multiplicative semigroup generated by \(z\). The main result of the paper shows that there exist embeddings \(J_{a,b}\subset A_2(k)\subset H_{a,b}\), where \(A_2(k)\) is a Weyl algebra.