A parametric family of subalgebras of the Weyl algebra I. Structure and automorphisms

Abstract

An Ore extension over a polynomial algebra F [ x ] \mathbb {F}[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A h \mathsf {A}_h generated by elements x , y x,y , which satisfy y x − x y = h yx-xy = h , where h ∈ F [ x ] h\in \mathbb {F}[x] . We investigate the family of algebras A h \mathsf {A}_h as h h ranges over all the polynomials in F [ x ] \mathbb {F}[x] . When h ≠ 0 h \neq 0 , the algebras A h \mathsf {A}_h are subalgebras of the Weyl algebra A 1 \mathsf {A}_1 and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of A h \mathsf {A}_h over arbitrary fields F \mathbb {F} and describe the invariants in A h \mathsf {A}_h under the automorphisms. We determine the center, normal elements, and height one prime ideals of A h \mathsf {A}_h , localizations and Ore sets for A h \mathsf {A}_h , and the Lie ideal [ A h , A h ] [\mathsf {A}_h,\mathsf {A}_h] . We also show that A h \mathsf {A}_h cannot be realized as a generalized Weyl algebra over F [ x ] \mathbb {F}[x] , except when h ∈ F h \in \mathbb {F} . In two sequels to this work, we completely describe the irreducible modules and derivations of A h \mathsf {A}_h over any field.