Algebra endomorphisms of Weyl algebras are determined by its restriction to the center

In this subsection we assume that $k$ is a field of positive char.

Lemma 10   The natural group homomorphism $\operatorname{Aut}_{\mbox{k}\operatorname{-alg}}(A_n(k)) \to \operatorname{Aut}_{\mbox{k}\operatorname{-alg}}(Z_n(k))$ (restriction map) is injective.

The uniqueness of operator $p$-th root for generators of $Z_n$. $\qedsymbol$

Note.

It is not surjective. For example, let $k$ be a field of odd characteristic and consider an algebra automorphism $\psi$ of $Z_1(k)$ given by

$$\psi(\xi^p)=\xi^p,\quad \psi(\eta^p)= -\eta^p.$$

Then by the uniqueness of the operator $p$-th root we see that the lift $\phi$ of $\psi$ should satisfy

$$\phi(\xi)=\xi, \quad \phi(\eta)=-\eta.$$

But this $\phi$ is not an algebra homomorphism.

Lemma 11   The restriction map

$$\operatorname{End}_{\mbox{k}\operatorname{-alg}} (A_n(k)) \to \operatorname{End}_{\mbox{k}\operatorname{-alg}} (Z_n(k))$$

is injective.

Let $\overline{A_n(k)}$ be a copy of $A_n(k)$ and consider two $k$-algebra homomorphisms

$$\phi^{(1)},\phi^{(2)}:\overline{A_n(k)} \to A_n(k).$$

Then we have $k$-algebra isomorphisms

$$\phi^{(1)}\otimes \operatorname{id}_{Z_n(k)}: \overline{A_n(k)}\otimes_{\overline{Z_n(k)}}Z_n(k) \to A_n(k)$$

where $\overline{Z_n(k)}$ is the center of $\overline{A_n(k)}$. Since these two isomorphisms coincide on the center, we deduce from the previous lemma that both maps coincide. $\qedsymbol$ ARRAY(0x8ec10fc)ARRAY(0x8ec10fc)