The group of automorphisms of the first Weyl algebra in prime characteristic and the restriction map

TL;DR

This paper characterizes the automorphism group of the first Weyl algebra over a perfect field of prime characteristic, establishing a monomorphism into a subgroup of automorphisms of its center and providing an explicit inverse formula.

Contribution

It explicitly describes the restriction map from automorphisms of the Weyl algebra to its center and characterizes its image, revealing structural properties and providing an explicit inverse formula.

Findings

The restriction map is a monomorphism with image equal to a specific subgroup of automorphisms of the center.

The map from automorphisms of the Weyl algebra to this subgroup is a monomorphism of infinite-dimensional algebraic groups, not an isomorphism.

An explicit formula for the inverse of the restriction map is derived using differential operators and the Frobenius map.

Abstract

Let $K$ be a {\em perfect} field of characteristic $p>0$, $A_1:=K< x, \der | \der x- x\der =1>$ be the first Weyl algebra and $Z:=K[X:=x^p, Y:=\der^p]$ be its centre. It is proved that $(i)$ the restriction map $\res :\Aut_K(A_1)\ra \Aut_K(Z), \s \mapsto \s|_Z$, is a monomorphism with $\im (\res) = \G :=\{\tau \in \Aut_K(Z) | \CJ (\tau) =1\}$ where $\CJ (\tau) $ is the Jacobian of $\tau$ (note that $\Aut_K(Z)=K^*\ltimes \G$ and if $K$ is {\em not} perfect then $\im (\res) \neq \G$); $(ii)$ the bijection $\res : \Aut_K(A_1) \ra \G$ is a monomorphism of infinite dimensional algebraic groups which is {\em not} an isomorphism (even if $K$ is algebraically closed); $(iii)$ an explicit formula for $\res^{-1}$ is found via differential operators $\CD (Z)$ on $Z$ and negative powers of the Fronenius map $F$. Proofs are based on the following (non-obvious) equality proved in the paper: $$ (\frac{d}{dx}+f)^p= (\frac{d}{dx})^p+\frac{d^{p-1}f}{dx^{p-1}}+f^p, f\in K[x].$$