The automorphism groups for a family of generalized Weyl algebras

Abstract

In this paper, we study a family of generalized Weyl algebras [Formula: see text] and their polynomial extensions. We will show that the algebra [Formula: see text] has a simple localization [Formula: see text] when none of [Formula: see text] and [Formula: see text] is a root of unity. As an application, we determine all the height-one prime ideals and the center for [Formula: see text], and prove that [Formula: see text] is cancellative. Then we will determine the automorphism group and solve the isomorphism problem for the generalized Weyl algebras [Formula: see text] and their polynomial extensions in the case where none of [Formula: see text] and [Formula: see text] is a root of unity. We will establish a quantum analogue of the Dixmier conjecture and compute the automorphism group for the simple localization [Formula: see text]. Moreover, we will completely determine the automorphism group for the algebra [Formula: see text] and its polynomial extension when [Formula: see text] and [Formula: see text].