Endomorphisms of right ideals of the Weyl algebra

Abstract:

Let $A = A(k)$ be the first Weyl algebra over an infinite field $k$, let $P$ be any noncyclic, projective right ideal of $A$ and set $S = \operatorname {End} (P)$. We prove that, as $k$-algebras, $S\not \cong A$. In contrast, there exists a noncyclic, projective right ideal $Q$ of $S$ such that $S \cong \operatorname {End} (Q)$. Thus, despite the fact that they are Morita equivalent, $S$ and $A$ have surprisingly different properties. For example, under the canonical maps, ${\operatorname {Aut} _k}(A) \cong {\operatorname {Pic} _k}(A) \cong {\operatorname {Pic} _k}(S)$. In contrast, ${\operatorname {Aut} _k}(S)$ has infinite index in ${\operatorname {Pic} _k}(S)$.