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

Original · EN

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 Aₕ generated by elements x,y, which satisfy yx-xy = h, where h∈ F[x]. We investigate the family of algebras Aₕ as h ranges over all the polynomials in F[x]. When h ≠ 0, these algebras are subalgebras of the Weyl algebra A₁ and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of Aₕ over arbitrary fields F and describe the invariants in Aₕ under the automorphisms. We determine the center, normal elements, and height one prime ideals of Aₕ, localizations and Ore sets for Aₕ, and the Lie ideal [Aₕ,Aₕ]. We also show that Aₕ cannot be realized as a generalized Weyl algebra over F[x], except when h ∈ F. In two sequels to this work, we completely describe the derivations and irreducible modules of Aₕ over any field.