Lifts of endomorphisms of Weyl algebras modulo $p^2$

TL;DR

This work characterizes when a $k$-algebra endomorphism $\varphi$ of the Weyl algebra $A_n(k)$ in positive characteristic $p$ lifts to an endomorphism of the Weyl algebra over Witt vectors $W_2(k)$. The lift exists precisely when the induced center map $\varphi_Z$ is a Poisson morphism, providing an obstruction-theoretic and cohomological framework that ties lifting to Poisson geometry on the center $Z$. By refining Tsuchimoto’s degree bounds, the authors show liftability (and hence Poisson preservation) under $\deg(\varphi) < p$, and they derive a differential-equation criterion involving auxiliary functions $f_i$ (or $\gamma_i$) that determines liftability. The paper further applies these results to obtain injectivity under the degree bound, and in birational cases to deduce flatness and automorphism properties; collectively, this advances the understanding of Weyl algebra automorphisms in positive characteristic and links lifting problems to de Rham–Poisson structures. The methods combine a concrete lifting construction with cohomological obstructions and $p$-power differential identities, yielding explicit criteria and broadening the scope of lifting phenomena beyond characteristic-zero analogues.

Abstract

Let $\varphi$ denote a $k$-algebra endomorphism of the $n$-th Weyl algebra $A_n(k)$ over a perfect field $k$ of positive characteristic $p$. We prove that $\varphi$ can be lifted to an endomorphism of the Weyl algebra $A_n(W_2(k))$ over the Witt vectors $W_2(k)$ of length two over $k$ if and only if $\varphi$ induces a Poisson morphism of the center of $A_n(k)$. Furthermore, we improve a result of Tsuchimoto, which enables us to conclude that these equivalent statements hold at least when ${\rm deg}(\varphi) < p$. In particular, we conclude that $\varphi$ is injective if ${\rm deg}(\varphi) < p$.

Theorems & Definitions (40)

• Lemma 1.1
• proof
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Theorem 2.4
• proof