Lifts of endomorphisms of Weyl algebras modulo p2

Let φ 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 φ 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 φ 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 deg ⁡ ( φ ) < p . In particular, we conclude that φ is injective if deg ⁡ ( φ ) < p .