Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple Lie Algebra in prime characteristic, we have the following sentence:

The sheaf $D_X$ of crystalline differential operators on a smooth variety $X$ over $k$ has a non-trivial center, canonically identified with the sheaf of functions on the Frobenius twist $T^∗X^{(1)}$ of the cotangent bundle. Moreover $D_X$ is an Azumaya algebra over $T^∗X^{(1)}$.

Instead of going through the general proof given, I only want to understand, in as simple a manner as possible, the situation when $X$ is the affine $n$-space over $k$. In this case, $D_X$ is simply the Weyl algebra and the Azumaya property, if I understand correctly, means that the quotient of the Weyl algebra by its centre is isomorphic to some matrix algebra over $k$. Is there a way to construct such a matrix algebra and a corresponding isomorphism to the quotient explicitly? Any help, even in the case of the affine line, would be highly appreciated.

P.S. If my understanding is incorrect, could you please point out the flaw(s) and how the question could be turned into something reasonable?

Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple Lie Algebra in prime characteristic, we have the following sentence:

The sheaf $D_X$ of crystalline differential operators on a smooth variety $X$ over $k$ has a non-trivial center, canonically identified with the sheaf of functions on the Frobenius twist $T^∗X^{(1)}$ of the cotangent bundle. Moreover $D_X$ is an Azumaya algebra over $T^∗X^{(1)}$.

Instead of going through the general proof given, I only want to understand, in as simple a manner as possible, the situation when $X$ is the affine $n$-space over $k$. In this case, $D_X$ is simply the Weyl algebra and the Azumaya property, if I understand correctly, means that the quotient of the Weyl algebra by its centre is isomorphic to some matrix algebra over $k$. Is there a way to construct such a matrix algebra and a corresponding isomorphism to the quotient explicitly? Any help, even in the case of the affine line, would be highly appreciated.

P.S. If my understanding is incorrect, could you please point out the flaw(s) and how the question could be turned into something reasonable?

Major Edit It has been pointed out that my understanding of the Azumaya property is incorrect. But my question remains the same: is there a direct way to prove the claim of the paper in the case when $X$ is the affine $n$-space (or even the affine line) over $k$.

Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Localisation of Modules for a semisimple Lie Algebra in prime characteristic, we have the following sentence:

The sheaf $D_X$ of crystalline differential operators on a smooth variety $X$ over $k$ has a non-trivial center, canonically identified with the sheaf of functions on the Frobenius twist $T^∗X^{(1)}$ of the cotangent bundle. Moreover $D_X$ is an Azumaya algebra over $T^∗X^{(1)}$.

Instead of going through the general proof given, I only want to understand, in as simple a manner as possible, the situation when $X$ is the affine $n$-space over $k$. In this case, $D_X$ is simply the Weyl algebra and the Azumaya property, if I understand correctly, means that the quotient of the Weyl algebra by its centre is isomorphic to some matrix algebra over $k$. Is there a way to construct such a matrix algebra and a corresponding isomorphism to the quotient explicitly? Any help, even in the case of the affine line, would be highly appreciated.

P.S. If my understanding is incorrect, could you please point out the flaw(s) and how the question could be turned into something reasonable?

Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple Lie Algebra in prime characteristic, we have the following sentence:

The sheaf $D_X$ of crystalline differential operators on a smooth variety $X$ over $k$ has a non-trivial center, canonically identified with the sheaf of functions on the Frobenius twist $T^∗X^{(1)}$ of the cotangent bundle. Moreover $D_X$ is an Azumaya algebra over $T^∗X^{(1)}$.

Instead of going through the general proof given, I only want to understand, in as simple a manner as possible, the situation when $X$ is the affine $n$-space over $k$. In this case, $D_X$ is simply the Weyl algebra and the Azumaya property, if I understand correctly, means that the quotient of the Weyl algebra by its centre is isomorphic to some matrix algebra over $k$. Is there a way to construct such a matrix algebra and a corresponding isomorphism to the quotient explicitly? Any help, even in the case of the affine line, would be highly appreciated.

P.S. If my understanding is incorrect, could you please point out the flaw(s) and how the question could be turned into something reasonable?