$k$-algebra, on $2n$ generators, defined as quotient of free algebra by 2-sided ideal generated by set, center & 2-sided ideals?

Question

Let $k$ be a field of $\text{char}\,k \neq 2$. For any $n \ge 1$, define a $k$-algebra $A_n(k)$, on $2n$ generators, as a quotient of the free algebra $k\langle x_1, \ldots, x_n, y_1, \ldots, y_n\rangle$ by the two-sided ideal generated by the set$$\{x_iy_j - y_jx_i - \delta_{ij},\,x_i x_j - x_jx_i, \,y_iy_j - y_jy_i, \, i , j = 1, \ldots, n\}, \quad \delta_{ij} := \begin{cases} 1 & \text{if }i = j \\ 0 & \text{if }i \neq j.\end{cases}$$I have a few questions.

• What is the center and all two-sided ideals of the algebra $A_n(k)$ in the case $\text{char}\,k = 0$?
• What is the center of the algebra $A_n(k)$ in the case $\text{char}\,k > 0$?

Answer 1

Your algebra $A_n(k)$ is the $n$th Weyl algebra. When the characterisitc of $k$ is $0$, it is simple and its center is trivial. When the characteristic of $k$ is $p>0$, the center is the subalgebra generated by the $p$th powers $x_i^p$ and $y_i^p$, for $i=1,2,\dots,n$, of the generators of $A_n(k)$.