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The Automorphism Groups for a family of Generalized Weyl Algebras

In this paper, we study a family of generalized Weyl algebras $\{\A\}$ and their polynomial extensions. We will show that the algebra $\A$ has a simple localization $\A_{\mathbb{S}}$ when none of $p$ and $q$ is a root of unity. As an application, we determine all the height-one prime ideals and the center for $\A$, and prove that $\A$ is cancellative. Then we will determine the automorphism group and solve the isomorphism problem for the generalized Weyl algebras $\A$ and their polynomial extensions in the case where none of $p$ and $q$ is a root of unity. We will establish a quantum analogue of the Dixmier conjecture and compute the automorphism group for the simple localization $(\mathcal{A}_{p}(1, 1, \K_{q}[s, t]))_{\mathbb{S}}$. Moreover, we will completely determine the automorphism group for the algebra $\mathcal{A}_{p}(1, 1, \K_{q}[s, t])$ and its polynomial extension when $p\neq 1$ and $q\neq 1$.