On the automorphism group of the first Weyl algebra

TL;DR

This paper investigates the automorphism group of the first Weyl algebra, proving that each Stafford subgroup equals its normalizer, thus deepening understanding of the algebra's automorphism structure.

Contribution

It establishes that all Stafford subgroups of the first Weyl algebra are equal to their normalizers, clarifying their role within the automorphism group.

Findings

Stafford subgroups are equal to their normalizers.

Characterization of when one Stafford subgroup contains another.

Enhanced understanding of automorphism group structure.

Abstract

Let $A_{1} := k [t, \partial ]$ be the first algebra over a field $k$ of characteristic zero. One can associate to each right ideal $I$ of $A_1$ its Stafford subgroup, which is a subgroup of $\Aut_k(A_1)$, the automorphism group of the ring $A_1$. In this article we show that each Stafford subgroup is equal to its normalizer. For that, we study when the Stafford subgroup of a right ideal of $A_1$ contains a given Stafford subgroup.