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 $h \neq 0$, the algebras $\mathsf {A}_h$ 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 irreducible modules and derivations of $\mathsf {A}_h$ over any field.