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

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 $h \neq 0$, 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.