A Parametric Family of Subalgebras of the Weyl Algebra I. Structure and Automorphisms

Abstract

An Ore extension over a polynomial algebra $\mathbb{F}[x]$ is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra ${\mathsf{A}}_h$ generated by elements $x,y$, which satisfy $yx-xy=h$, where $h\in \mathbb{F}[x]$. We investigate the family of algebras ${\mathsf{A}}_h$ as $h$ ranges over all the polynomials in $\mathbb{F}[x]$. When $h\ne 0$, these algebras are subalgebras of the Weyl algebra ${\mathsf{A}}_1$ and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of ${\mathsf{A}}_h$ over arbitrary fields $\mathbb{F}$ and describe the invariants in ${\mathsf{A}}_h$ under the automorphisms. We determine the center, normal elements, and height one prime ideals of ${\mathsf{A}}_h$, localizations and Ore sets for ${\mathsf{A}}_h$, and the Lie ideal $[{\mathsf{A}}_h,{\mathsf{A}}_h]$. We also show that ${\mathsf{A}}_h$ cannot be realized as a generalized Weyl algebra over $\mathbb{F}[x]$, except when $h\in \mathbb{F}$. In two sequels to this work, we completely describe the derivations and irreducible modules of ${\mathsf{A}}_h$ over any field.

36 pages, A few minor changes and references updated