A Parametric Family of Subalgebras of the Weyl Algebra II. Irreducible Modules

Abstract

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