The quotient field of the Weyl algebra

Let $k$ be a field of characteristic $p>0$. There is a nice ``quotient field of the Weyl algebra $A_n(k)$.

Lemma 6   Let $K_n(k)$ be the quotient field of the center $Z_n(k)$ of the Weyl algebra $Z_n(k)$. Then the following statements hold.

1. $D_n(k)=A_n(k)\otimes_{Z_n(k)} K_n(k)$ is a skew field.
2. $\dim_{K_n(k)} D=p^{2n}$.
3. $M_{p^n}(L_n(k))$ may be regarded as a $p^{2n}$-dimensional vector space over $D$ with a basis $\{T^IU^J ; \vert\vert I\vert\vert _{\ell^\infty}\leq p-1, \vert\vert J\vert\vert _{\ell^\infty}\leq p-1\}$.

(1) $D_n(k)$ is a simple algebra with no zero-divisor except for 0. Then we use Wedderburn's structure theorem.

(2),(3): we may easily see that

$$\dim_{K_n(k)} D_n(k)\leq p^{2n} , \quad \dim_{D_n(k)} M_p(L_n(k))\leq p^{2n}.$$

On the other hand we have

$$\dim_{K_n(k)}M_p(L_n(k))=p^{4n}$$

$\qedsymbol$ ARRAY(0x8e89d54)