The group of order preserving automorphisms of the ring of differential operators on Laurent polynomial algebra in prime characteristic

TL;DR

This paper characterizes the group of order-preserving automorphisms of the ring of differential operators on Laurent polynomial algebras in prime characteristic, revealing its structure as a skew direct product involving p-adic integers.

Contribution

It explicitly determines the structure of the automorphism group of differential operators on Laurent polynomial algebras in prime characteristic, including an explicit description.

Findings

The automorphism group is isomorphic to a skew direct product of alp^n and \u00a7Aut_K(L_n)

The automorphism group of differential operators on polynomial algebras is isomorphic to (K[x_1,...,x_n])

The group structure is explicitly described in terms of p-adic integers and algebra automorphisms.

Abstract

Let $K$ be a field of characteristic $p>0$. It is proved that the group $\Aut_{ord}(\CD (L_n))$ of order preserving automorphisms of the ring $\CD (L_n)$ of differential operators on a Laurent polynomial algebra $L_n:= K[x_1^{\pm 1}, ..., x_n^{\pm 1}]$ is isomorphic to a skew direct product of groups $\Zp^n \rtimes \Aut_K(L_n)$ where $\Zp$ is the ring of $p$-adic integers. Moreover, the group $\Aut_{ord}(\CD (L_n))$ is found explicitly. Similarly, $\Aut_{ord}(\CDPn)\simeq \Aut_K(P_n)$ where $P_n: =K[x_1, ..., x_n]$ is a polynomial algebra.