Notations

All rings are assumed to be unital, associative. All homomorphisms are assumed to be unital. $\mathbb{N}=\{0,1,2,3,\dots\}$

For any ring $R$ and for any positive integer $l$, we put

$$M_l(R)=($$the set of $l×#times;l$-matrix with coefficients in $R$.$$)$$

For any ring $k$, we denote by $A_n(k)$ the Weyl algebra.

$$A_n(k)=k\langle \xi_1,\xi_2,\dots,x_n,\eta_1,\eta_2,\dots,\eta_n \rangle /(\eta_j\xi_i-\xi_i\eta_j-\delta_{ij}; 1\leq i,j\leq n)$$

where $\delta_{ij}$ is the Kronecker's delta.

If $k$ is a field with characteristic $p$, then we further employ the following notations.

$Z_n(k)=Z(A_n(k))$     the center of $A_n(k)$ (Later we prove that $Z_n(k)$ is actually equal to $k[\xi_1^p,\xi_2^p,\dots,\xi_n^p,\eta_1^p,\eta_2^p,\dots,\eta_i^p])$.

$T_i=(\xi_i^p)^{1/p}$

$U_i=(\eta_i^p)^{1/p}$

$S_n(k)=Z_n(k)^{1/p}=k^{1/p}[T_1,T_2,\dots,T_n,U_1,U_2,\dots,U_n]$. ($p$-th roots are taken in the usual sense of commutative algebra. $S_n(k)$ is a commutative algebra over $Z_n(k)$.)

$K_n(k)=Q(Z_n(k))$, the quotient field of $Z_n(k)$.

$D_n(k)=A_n(k)\otimes_{Z_n(k)} K_n(k)$

$L_n(k)=Q(S_n(k))$.

$B_n(k)=A_n(k)\otimes_{Z_n(k)} S_n(k)$ ( $\cong M_{p^n}(S_n(k))$)

$\mathcal V_n(k)=\oplus_{i=1}^{p^n}S_n(k)$

$E_n(k)=A_n(k)\otimes_{Z_n(k)} L_n(k)$ ( $\cong M_{p^n}(L_n(k))$)

$\alpha_i=\xi_i-T_i$

$\beta_i=\eta_i-U_i$

$\overline{A_n}$ : a copy of $A_n$

(Identified with the image of $\phi$ when $\phi$ is injective).

$\overline{Z_n}=Z(\overline{A_n})$ : a copy of $Z_n$

$\overline{S_n}$ : a copy of $S_n$ (extension of $\overline{Z_n}$)

Similarly we use a notation $\overline{\xi_i}, \overline{\eta_i}, \overline{T_i}, \overline{U_i}$ to indicate a copy of non-bar counterparts.

For a matrix $x$, we denote by $\lambda(x),\rho(x),\operatorname{ad}(x)$ the left action, the right action, and the adjoint action by $x$, respectively.

$$\lambda(x)y=xy, \quad \rho(x)y=yx , \quad ad(x)y=xy-yx$$